Real Functions

September 25, 2011

1 Introduction

A function over D, being D is a rule that for every number x of D ⊂ ℜ associates only one number of . It is represented in the form f : D . ℜ → ℜ y is called image of x and it is written as y = f(x). It is understood by domain of the function f the set: Domf = x : y with y = f(x) { ∈ℜ ∃ ∈ℜ } While the image of the function is deﬁned as the set: Imf = y : x with y = f(x) { ∈ℜ ∃ ∈ℜ } The graphic of the function f is deﬁned as the subset of 2: ℜ G(f)= (x,y) 2 : y = f(x) { ∈ℜ } 2 Operations with functions

Given two functions f, g : , we deﬁne the following operations: ℜ←ℜ sum of two functions: • (f + g)(x)= f(x)+ g(x) The domain of f + g is the intersection of the domains Dom f Dom g. ∩ product of two functions: • (f g)(x)= f(x) g(x) · · The domain of f + g is the intersection of the domains Dom f Dom g. ∩ Composition of two functions: • (g f)(x)= g(f(x)) ◦ This function exists for the x so that f(x) Dom g, for every x ∈ ℜ ∈ ∈ ℜ so that Imf Dom g. ⊂

1 2.1 Kinds of functions Given a function f : , it is said that it is an even function when it • ℜ→ℜ veriﬁes that:

f( x)= f(x), x Domf − ∀ ∈ . This function would present symmetry with respect to OY axis. and it is odd when it veriﬁes that

f( x)= f(x), x Domf − − ∀ ∈ . This function would present symmetry with respect to the origin. Given a function f : , it is said to be periodic if it veriﬁes that • ℜ→ℜ it exists a real number T that

f(x + T )= f(x), x Dom f ∀ ∈ The minimum number T that veriﬁes the previous condition is called period of the function.

3 Elemental functions

A constant function has the form f(x)= c, being c any real number. This • functions transforms every real number in the number c. Its domain is ℜ and the image is c. A lineal function has the form f(x) = mx, being m a real constant • diﬀerent from zero. They verify that f(αx1 + βx2) = αf(x1) + βf(x2). This function has as domain and image every real number. Its graphic is a straight line that crosses the origin with a slope m. If m> 0 the function is strictly increasing. If m< 0 the function is strictly decreasing. Aﬃne functions have the form f(x) = mx + n, where m,n and • ∈ ℜ m = 0. This functions have as domain and image all the real numbers. Its 6 graphic is a straight line of equation y = mx + n, that crosses the point (0,n) with a slope m. Potential functions have the form f(x) = xn, where n . This • ∈ ℜ functions have as domain all the real numbers. Their image depends on the parity of n. If n is even, the image is all positive real numbers [; + inf). If n is odd, then the image is every number in . ℜ

2 Dirichlet function. It is defined as follows: • 0 ifx Q, f(x)= ∈ (1) 1 ifx/ Q. ∈ Its domain is every real number, rationals and irrationals. Its image is composed only by the numbers 0, 1 . Every rational has as image 0 and { } every irrational has as image 1. The sign function. It is defined as follows: • 1 if x> 0, sign(x)=  0 if x = 0, (2)  1 if x< 0. −  Its domain is every real number. Its image is composed only by the numbers 0, 1, 1 { − } The absolute value function. It is defined as follows: • x ifx> 0, x = (3) | | x ifx< 0. − 3.1 Conic functions Circumference. The equation of a circumference with centre at (0,0) and • radius r os x2 + y2 = r2. Isolating y,

y = r2 x2 ±p − that is not a function (for one value of x we get two values of y). We can consider two functions instead,

y = r2 x2 p − y = r2 x2 −p − Their domain is [-r;r] and their images are [0;r] and [-r;0] respectively. Ellipse. The equation of an ellipse with centre at (0,0) and semi-axes • x2 y2 a, b > 0 a2 + b2 = 1. Isolating y, b y = a2 x2 ±ap − that is not a function. We can consider two functions instead,

3 b y = a2 x2 ap − b y = a2 x2 −ap − Their domain is [-a;a] and their images are [0;b] and [-b;0] respectively. Hyperbola. The equation of an ellipse with centre at (0,0) and semi-axes • x2 y2 a, b > 0 2 2 = 1. Isolating y, a − b b y = x2 a2 ±ap − that is not a function. We can consider two functions instead,

b y = x2 a2 ap − b y = x2 a2 −ap − Their domain is ( ; a] [a; ) and their images are [0; ) ( ; 0] −∞ ∪ ∞ ∞ ∪ −∞ respectively. Parabola. The most used parabolas are those that have as equations • y = ax2 +bx+c with a, b, c (their domain is ) and y2 = px with p . ∈ℜ ℜ ∈ℜ Isolating y, b y = √px ±a that is not a function. We can consider two functions instead,

b y = √px a b y = √px −a The domain for these two functions is [0; + ) if p > 0 and ( ; 0] if ∞ −∞ p< 0.

3.2 Trigonometric functions Sine, Cosine and Tangent. This functions are periodic and not inyective • so we have to restrict their domain to an interval where they are inyective. For function y = sin(x) we consider the interval [ π ; π ] and its image is − 2 2 [ 1; 1]. Its inverse function is y = arcsin(x), its domain is the image of − y = sin(x) and is image is [ π ; π ]. For y = cos(x) we choose the interval − 2 2

4 [0; π] where it is inyective and strictly decreasing. Its image is [ 1; 1]. Its − inverse function is the function y = arccos(x) with domain [ 1; 1] and image − [0; π] and also strictly decreasing. For the function y = tg(x) we consider the interval [ π ; π ] (the same as the sin function). Where it is inyective and − 2 2 increasing. Its inverse function y = arctg(x) transforms into this interval. ℜ Trigonometric functions verify important and fundamental relations:

cos2(x)+ sin2(x) = 1 sin(x) tg(x)= cos(x) 1 1+ tg2(x)= cos2(x) Relations of addition:

sin(x y) = sin(x) cos(y) cos(x) sin(y) ± ± cos(x y) = cos(x) cos(y) sin(x) sin(y) ± ∓ tg(x) tg(y) tg(x y)= ± ± 1 tg(x)tg(y) ∓ Relations of double arc:

sin(2x) = 2sin(x) cos(x) cos(2x) = cos2(x) sin2(x) − Relations of half arc:

x 1 cos(x) sin( )= − 2 ±r 2 x 1 + cos(x) cos( )= 2 ±r 2 Reciprocal trigonometric functions: Secant. • 1 sec(x)= cos(x) Cosecant. • 1 cosec(x)= sin(x) Cotangent. • 1 cotg(x)= tg(x)

5 3.3 Exponential functions The exponential function of base a is deﬁned as f(x) = ax with a > 0. In the case a = 1 then the function is constant and equal to 1 f(x) = 1. Its graphic is diﬀerent if 0 1. 1 n The number e is the limit of the series (1 + n ) , 1 e = lim(1 + )n n n 3.4 Logarithmic functions The logarithmic function of base a, being a > 0 and a = 1, is given by 6 y = loga(x). Exponential and logarithmic functions are inverse functions. The most important properties of the logarithms are the following:

loga(xy)= loga(x)+ loga(y) x log ( )= log (x) log (y) a y a − a n loga(x )= nloga(x) When a is not speciﬁed a = 10 and when a = e we write y = ln(x) and we call it Neperian Logarithm.

3.5 Hyperbolic functions Hyperbolic functions are based on the exponential function: ex e−x sh(x)= − 2 ex + e−x ch(x)= 2 ex e−x th(x)= − ex + e−x These functions verify the relations:

ch(x)2 sh(x)2 = 1 − sh(x) th(x)= ch(x) 1 1 th(x)2 = − ch(x)2 The argument functions of hyperbolic functions are also interesting: argsh(x), argch(x), argtgh(x)

6 4 Representation of functions

4.1 Asymptotes An asymptote is a straight line such that the distance between the curve and the line approaches zero as they tend to one speciﬁc value. They can be horizontal, vertical or obliques. The line of equation x = a is a vertical asymptote of the function f if lim f(x)= inf when x a, when x a+ or when x a−. It means that ± → → → when the function f approximates to the point a, at least from one lateral, the values of the function goes to inf. ± The straight line of equation y = k is an horizontal asymptote of function f if

lim f(x)= k, or lim f(x)= k with k x→+∞ x→−∞ ∈ℜ This means that the distance between the asymptote and the function goes to zeros as x goes to inﬁnity. There can be two horizontal asymptotes, one for + and another for , that can be the same line, there can ∞ −∞ exist an asymptote for only one of this values or there can be no horizontal asymptotes. The straight line of equation y = mx + n is said to be an oblique asymptote of the function f if

x f(x) y 0, →±∞⇒ − → what is to say, the distance between the function and the asymptote goes to zero when x goes to plus or minus inﬁnity. The value of the slope of the asymptote is obtained as follows,

f(x) m = lim x→±∞ x and m must be ﬁnite and diﬀerent to zero. The value of n is given by,

n = lim f(x) mx x→±∞ − There can exist only two oblique asymptotes, one for + and another ∞ for that, in some cases, can be the same line. What is more, in total −∞ there can be only two asymptotes. Thus, if a function has two horizontal asymptotes we should not have to look for oblique asymptotes or vice-versa.

7 4.2 Study of the monotonicity If a function is derivable at a point, the fact that its derivative is positive or negative gives us an idea about its monotonicity (increasing or decreasing) near the point.

Theorem 1 Let f be derivable in an interval I, we have that: 1. If f ′(x) > 0, x I, then f is strictly increasing in I, ′ ∀ ∈ 2. If f (x) < 0, x I, then f is strictly decreasing in I, ′ ∀ ∈ 3. If f (x) = 0, x I, then f is constant in I. ∀ ∈ 4.3 Local extrema

Theorem 2 If f has a local relative extrema at x0 and f(x) is derivable ′ at x0, then f (x0) = 0. These points x0 are called critical points. Even though, a function can be derivable at a point and be zero its derivative at this point and not have an extrema at this point (but there would be a change of convexity or inﬂexion point).

′ Theorem 3 If a function f is that f (x0) = 0 and it exists a δ that 1. f ′(x ) < 0 in (x δ; x ) and f ′(x ) > 0 in (x ; x + δ), then x is a local 0 0 − 0 0 0 0 0 minima. 2. f ′(x ) > 0 in (x δ; x ) and f ′(x ) < 0 in (x ; x + δ), then x is a local 0 0 − 0 0 0 0 0 maxima.

′ Theorem 4 Given a function f, twice derivable in x0, being f (x0) = 0, we can aﬃrm that ′′ ′′ 1. if f (x0) > 0, then f(x0) is a local minimum, 2. if f (x0) < 0, then f(x0) is a local maximum. ′′ In the case of functions that verify f (x0) = 0, this criteria does not decide the existence of local extrema, so we should refer to the previous theorem.

4.3.1 Absolute extrema

It is said that f(x) has at the point x0 an absolute minimum if it verifies that f(x ) f(x) for all the values of x inside the domain of f. It has an 0 ≤ absolute maximum if it is verified that f(x ) f(x) for all the values inside 0 ≥ the domain of f. In the case that a function is defined in an closed interval [a; b], the absolute extrema of the function can be reached at the extrema of the interval,

8 what is to say, at a or b, at the critical points or at points where the function has no derivative.

4.4 Study of the convexity Theorem 5 Let f be derivable in an interval I, we have that: 1. If f ′′(x) > 0, x I, then f is convex, ′ ∀ ∈ 2. If f (x) < 0, x I, then f is concave, ′ ∀ ∈ 3. If f (x) = 0, x I, then f is an inﬂexion point. ∀ ∈ 5 Derivatives of elemental functions

5.1 Derivation rules

′ ′ ′ (f + g) = f + g ′ ′ ′ (f g) = f g − ′ ′ − ′ (fg) = f g + fg ′ f f ′g gf ′ = − g g2 ′ ′ (αf) = αf ′ ′ ′ (g f) (x) = g (f(x)) f (x) ◦ · − ′ 1 (f 1) (x) = f ′(f −1(x))

5.2 Derivatives of elemental functions

′ − P otential f(x) = xa, f (x)= axa 1 ′ 1 − 1 Root square f(x) = x1/2 = √x, f (x)= x 1/2 = 2 2√x ′ 1 1 − 1 1/n n n 1 f(x) = x = √x, f (x)= x = n n n √xn−1 ′ 1 Logarithm f(x) = lnx, f (x)= x ′ 1 f(x) = log x, f (x)= log e a x a ′ Exponential f(x) = ex, f (x)= ex ′ f(x) = ax, f (x)= axln(a)

9 ′ − ′ ′ P otential exponential f(x) = uv, f (x)= vuv 1u + uvln(u) v − ′ · T rigonometricfunctions f(x) = cos(x), f (x)= sin(x) ′ − f(x) = sin(x), f (x) = cos(x) sin(x) ′ 1 f(x)= tg(x) = , f (x)= =1+ tg2(x) cos(x) cos2(x) cos(x) ′ 1 f(x)= cotg(x) = , f (x)= − = (1 + cotg2(x)) sin(x) sin2(x) − ′ 1 Inverse trigonometric functions f(x) = arccos(x), f (x)= −√1 x2 − ′ 1 f(x) = arcsin(x), f (x)= √1 x2 − ′ 1 f(x) = arctg(x), f (x)= 1+ x2 ′ Hyperbolic functions f(x) = sh(x), f (x)= ch(x) ′ f(x) = ch(x), f (x)= sh(x) ′ 1 f(x) = th(x), f (x)= = 1 th2(x) ch2(x) − ′ 1 f(x) = argch(x), f (x)= √1 x2 − ′ 1 f(x) = argsh(x), f (x)= √1+ x2 ′ 1 f(x) = argth(x), f (x)= 1 x2 −

6 Limits of functions

We say that the function f has limit L when x tends to a, if for every ǫ> 0, it exists δ > 0 so that for all the x that 0 < x a < δ then f(x) L <ǫ. | − | | − | 6.1 Properties of limits The principal properties of the limit of a function are the following: 1. The limit of a function at a point, if it exists, it is unique.

lim f(x)= L1 y lim f(x)= L2 then L1 = L2 x→a x→a

2. If limx→a f(x)= L1 y limx→a g(x)= L2, then (a) limx→a (f(x)+ g(x)) = L1 + L2

10 (b) lim → (αf(x)) = αL , α x a 1 ∈ℜ (c) limx→a (f(x)g(x)) = L1L2 1 1 (d) lim → = if L = 0 x a f(x) L2 2 6 f(x) L1 (e) lim → = if L = 0 x a g(x) L2 2 6 3. Sandwich rule: if x (a; b) f (x) f(x) f (x) and lim → f (x) = ∀ ∈ 1 ≤ ≤ 2 x c 1 lim → f (x)= L and c (a; b), then lim → f(x)= L. x c 2 ∈ x c 4. If f and g are functions that near the point a are f(x)= g(x), but in the point a, so that it exists limx→a f(x)= L, then it also exist limx→a g(x) and limx→a f(x) = limx→a g(x)= L

6.2 Lateral limits In the deﬁnition of limit we take values of x near a at both sides of a. It can happen that the limit exists only when we approximate a from one side. We say that the limit of f(x) when x goes to a from the left is L, and we write limx→a− f(x)= L when,

ǫ> 0, δ > 0 : 0 < x a <δ and x

ǫ> 0, δ > 0 : 0 < x a <δ and x>a then f(x) L <ǫ ∀ ∃ | − | | − | . In order to assure the existence of the limit of f(x) at the point a, lateral limits must exist and be equal.

6.3 Calculation of limits: Indeterminations Apart from the indetermination of the form k , with k = 0, that makes to 0 6 obtain lateral limits in order to decide if the limit exists or not, we can ﬁnd 7 indeterminations more that are represented as follows:

0 ∞ , ∞ , [ ] , [0 ] , 00 , 0 , [1 ] 0 ∞−∞ ·∞ ∞ h∞i

11 7 Continuity at a point

We say that a function is continuous at a point when the function is deﬁned at that point, the limit at that point exists and both values are equal:

1) f(a), ∃ f iscontinuous at a when =  2) limx→a f(x), (4)  ∃ 3) limx→a f(x)= f(a). This deﬁnition can be written also as

ǫ> 0, δ > 0 that if x a < δ then f(x) f(a) <ǫ ∀ ∃ | − | | − | . A function f is continuous from the right at the point a if it exists the value f(a) and it is equal to the limit of f from the right. A function f is continuous from the left at the point a if it exists the value f(a) and it is equal to the limit of f from the left. f is continuous at the point a if it is continuous from the left and from the right. Continuous functions have analogue properties to limits: if f and g are continuous at a if: 1. f + g is continuous at a, 2. f g is continuous at a, · 3. αf is continuous at a, being α R, ∈ 4. 1 is continuous at a, if g(a) = 0 g 6 5. f is continuous at a, if g(a) = 0 g 6 6. If f is continuous at a and g is continuous at b = f(a), then the function g f is continuous at a. ◦ 7.1 Discontinuities When a function does not fit any of the three conditions of the definition, the function is said not to be continuous at a. We say that the discontinuity is avoidable if the limit and the value of the function are not equal or if the function is not defined even though the limit exists. In that case, the function can be redefined and the discontinuity has been avoided. The discontinuity is called jump discontinuity when the limit does not exist but the lateral limits do. The jump can be finite when both lateral limits are finite, or infinite when one of the lateral limits is infinite. The discontinuity is said to be essential when one or both lateral limits do not exist.

12 8 Continuous functions theorems

It has special interest because of its applications, the study of the continuity of a function in an interval. If we are talking about an opened interval, the deﬁnition is:

f is continuous in (a, b) when x (a, b) : f is continuous in x. ∀ ∈ If we are talking about a closed interval we say that,

f is continuous in (a, b), f iscontinuous in [a, b] when  f is continuous from the right at a,  f is continuous from the left at b.  (5) In the case that we have functions defined in semiopened intervals or defined at unions of intervals, the same criteria will be used, what is to say, the points at the frontier of the set mus have lateral continuity. If the function is defined at an isolated point, the function must be continuous at this point.