Functions, Limits, Continuity (part 3)

Continuity.

Continuity: an Intuitive Definition

Intuitively, a function is continuous if you can draw it without lifting your pen from the paper. In the diagram below, the function on the left is continuous throughout, but the function on the right is not. It is "discontinuous" at x = c.

Figure %: A continuous (left) and discontinuous (right) function

Continuity: A Formal Definition

Formally, a function is continuous at a point x = c if the (standard two-sided) limit exists there and is equal to the value of the function at c. In other words, if

f (x) = f (c)

In order to show that

f (x) = f (c)

you need to show that

f (x)

exists, and

f (x)

exists, and that they are both equal to f (c).
A function is considered continuous if it is continuous at all points in its domain.

Some Important Continuous Functions

You might recognize that the formal requirement of continuity, i.e. that

f (x) = f (c)

is a property of polynomial functions. Thus, all polynomial functions are continuous. The following functions are always continuous, and you should be aware of them:
1. Polynomial Functions
2. Rational Functions, wherever the denominator is nonzero.
3. sin(x) and cos(x)
4. The sum, difference, product, and quotient (as long as the denominator is nonzero) of two continuous functions is continuous.

Demonstrating the Continuity of a Piecewise Function

One problem that you might have to deal with is using the formal definition of continuity to determine whether a piecewise-defined function is continuous.

Example: is f a continuous function?

f (x) =

Solution:

For a function to be continuous, it must be continuous at every point in its domain. The obvious point for us to be worried about here is the point where the definition of f changes, i.e. at x = 2. At places other than at x = 2 , f is defined by polynomial functions, which we know are continuous. It is the point where these two continuous functions meet that concerns us.
Therefore, to prove that f is a continuous function, we must prove that it is continuous at x = 2. In other words, we must show that

f (x) = f (2)

First let's see if f (x) exists by checking the left-hand and right-hand limits. As x approaches 2 from the left, f (x) is defined by the function 2x ² - 2 , so

f (x) = 2x 2-2 = 2(2)2 - 2 = 6

As x approaches 2 from the right, f (x) is defined by the function 5x - 4 , so

f (x) = 5x-4 = 5(2) - 4 = 6

Since

f (x) = f (x) = 6,

we can say that

f (x) = 6.

At x = 2 , f (x) is defined by 2x ² - 2 , so f (2) = 2(2)² - 2 = 6. Now we have shown that

f (x) = f (2)

which shows that f (x) is continuous at x = 2. Since f (x) is also continuous when x does not equal 2, f (x) is a continuous function. Below is a graph of f (x) to help you visualize what we have just done:

Figure %: A continuous (left) and discontinuous (right) function

The Intermediate Value Theorem

The intermediate value theorem says that if f is continuous on the closed interval [a, b] , then f attains each of the values between f (a) and f (b) at least once on the open interval (a, b).
A real-life example may help here. The temperature at various times of the day is a good example of a continuous function. Let's say that at 6am, it is 46 degrees outside, and by noon, it is 67 degrees. By the intermediate value theorem, at some time between 6am and noon, the temperature outside must have been exactly 51.7 degrees. We can pick any value between 46 and 67 and be confident that that exact temperature was attained sometime between 6am and noon.
We can also understand the intermediate value theorem graphically. Below is a graph of a function f that is continuous on [a.b]. Note that every value between f (a) and f (b) is attained somewhere on the interval (a, b).

Figure %: Each value between f (a) and f (b) is attained somewhere on the interval (a, b)