Continuity and One-Sided Limits
DIFFERENTIAL CALCULUS
COURSE /
3.4. CONTINUITY OF A FUNCTION
3.1.1 Define the continuity of a function at a point
3.1.2 Determine if a function is continuous or discontinuous at a given point from its algebraic expression or its graph
Let f be a function and . We say that f is continuous at if and only if . Knowing that if it can be said that , so the condition of a function to be continuous is .
If the function f is not continuous at x=c, then f is said to have a discontinuity at c. There are two types of discontinuties.
- Removable discontinuity
- Nonremovable discontinuity
o Jump discontinuity
o End of domain
o Infinite discontinuity
1) Removable discontinuity is the discontinuity at a point which could be removed by defining the function at just that one point. This often appears as a graph with a hole in it(gap). A removable discontinuity it appears when
An example of a removable discontinuity could be considered the function in the point x=2.
/ f(2)=dne
so because and f(2)=dne, the function is not continuous in x=2 and it has a removable discontinuity.
in x=2
/ so
f(2)=2
so because
and
f(2)=2, the function is not continuous in x=2 and it has a removable discontinuity.
2) Nonremovable discontinuity is the discontinuity at a point which could not be removed at just that point. There are three types of nonremovable discontinuities:
2a) Jump discontinuity is usually caused by a piecewise-defined function whose pieces don't really meet together.
In other words and and
Example1:
/ .
because the limit from the left is not equal with the limit from the right
and f(1)= -1
The function is not continuous in the x=1 and it has a jump discontinuity
2b) End of domain is when the function has a limit from one side, and from the other side does not exist.
In other words
and
OR
and /
An example of end of domain discontinuity is:
/ so
The value of the function in x=2 is f(2)=1
Because the limit does not exist from the left and the limit from the right is 1, the type of discontinuity is end of domain.
2c) Infinite discontinuity is defined where the jump is to the infinity
In other words
and OR and
in x=0
/ Because the limit from the left is not equal with the limit from the right, the function is not continuous in x=0 and the type of discontinuity is infinite discontinuity.
Discontinuity at a point.
A function has a discontinuity at a point x=c if and only if
Discontinuity on an interval
A function f is continuous on the closed interval if it is continuous on the open interval and and
Example1:
Given the following piecewise function
a) Analyze the function in each step point and decide the continuity or the type of discontinuity( if it exists)
b) Decide if the function is continuous in its entirety
c) Graph the function
d) Range of the function
a) To analyze the function in its step-points, it is needed to check the continuity in these step points (Step-points are the points of a piecewise function, where a component function is finishing, and other component function is starting). In our case, there are two step-points where is needed to check the continuity, at x=-2 and x=2.
In x=-2
From the left of x=-2, and from the right, . Because the limit from the left and the limit from the right has the same value, it can be decided that
The value of the function at x=-2 is .
Because of it can be concluded that the piecewise function is continuous in x=-2
In x=2
From the left of x=2, and from the right, . Because the limit from the left and the limit from the right has different values, in this point there is a discontinuity. So, the limit .
The value of the function at x=2 is .
The limit because of the limit from the left has a different value than the limit from the right and it can be concluded that the piecewise function has a jump discontinuity at x=-2
b) To decide the continuity in its entirety, it is needed to check each function´s continuity. The first function, is continuous in its entirety of domain, the second function also is continuous in its entirety of domain and the third function is also continuous in its domain.
/ d)
The range of the function is: