205 Lecture 2.6. Continuity

The idea of a continuous function is pretty intuitive. A continuous function has no holes or breaks. You can draw it without lifting your pencil off the page. However, modern functions can be impossible to draw (google the Dirichlet function or Thomae’s function), so the intuition breaks down sometimes. The mathematical definition of continuity is simple and clear.

Definition. A function f(x) is continuous at x = c if lim f(x) = f(c). x→c

Note that the equation in the definition is false if either f(c) does not exist or the limit does not exist, so in those cases the function is not continuous at c. Pictures are valuable for the concept of continuity.

5 5

2.5 2.5

-10 -7.5 -5 -2.5 0 2.5 5 7.5 10-10 -7.5 -5 -2.5 0 2.5 5 7.5 10

-2.5 -2.5

-5 -5

Polynomials and sine waves are continuous at all x.

y

(3, 1) x

Here f(3) is undefined, so f is not continuous at x = 3. y

(3, 1) x

Here, f(3) = 2 6= 1 = lim f(x), so f is not continuous at x = 3. x→3

y

x

(3, −1)

Here lim f(x) d.n.e. (the one-sided limits are not equal), so f is not continuous at 3. x→3

For the most part, STEM majors will be dealing with everywhere-continuous functions, or at least functions that are continuous on their domains. Continuity becomes increasingly important in math classes beyond Calc 1, because it plays a big role in the theoretical framework of calculus and the real numbers. However, all of the basic functions you learn about in precalculus are continuous on their domains, except piecewise functions, which are specifically designed to explore the concept√ of continuity. Rational functions are not continuous at vertical asymptotes, and radicals like x are continuous at every point interior to their domain and have one-sided continuity at points on the boundary of their domain. When I say “if you can plug in, do it” when evaluating a limit, it is because the functions involved are continuous. In fact, that is exactly what continuity tells us, i.e., it tells us when you can plug in to find a limit. Once again, by definition, f is continuous at x = c if lim f(x) = f(c). x→c When you replace regular limits with one-sided limits in the definition of continuity, you get one- sided continuity. In the figure above, lim f(x) = −1 and limx→3+ f(x) = 0. Note that f(3) = 0. x→3−

y

x

(3, −1)

Here f is continuous from the right at 3.

Therefore, limx→3+ f(x) = f(3), meaning f(x) is continuous from the right at x = 3. Since −1 6= 0, we know lim f(x) 6= f(3), so f is not continuous from the left at x = 3. The only discontinuity is x→3− at x = 3, so we can list intervals where f is continuous. We write “f is continuous on (−∞, 3) and on [3, ∞).” Note the bracket vs parenthesis. Writing “f is continuous on [3, ∞)” does not mean that f is continuous at 3. This confuses students, because 3 is in that interval. We do this because we want to indicate one-sided continuity, and the notation is convenient. When we write “f is continuous on [a, b],” we do not mean at a or at b. Rather, we mean that f is continuous at every point in the open interval (a, b), f is continuous from the left at b, and f is continuous from the right at a.