Recall that a function is continuous at a given point if it satisfies the three conditions shown here.

We can prove that if a function is differentiable at a given point, then the function is also continuous at that point.

We begin the proof with a statement of an alternative form of the definition of the derivative. Since we start with the assumption that f′ exists, we have only to prove the second part of the statement. But, f(a) must exist for the limit to have meaning and exist. Thus we can conclude that f(a) exists, and the first condition for continuity is satisfied.

Now, we rewrite the numerator of the limit, f(x) - f(a), by multiplying and dividing it by x - a. After rearranging the terms, only f(x) remains on the left.

Next, we take the limit of both sides. Using limit laws, we can represent the limit term on the right side of the equation as the product of two limits plus a third limit.

We recognize the first limit term on the right as the derivative f ′(x). The second limit term evaluates to zero. The third limit term is a constant, and thus is simply f(a).

We see that the limit of f(x) as x approaches a is just f(a). This satisfies the second and third conditions for continuity. Therefore, a function is continuous at a point x = a if it is differentiable at that point.

Next let's consider the piecewise function shown here. The graph of the function has a "kink" in it at x = 3.

What can we conclude about the function's continuity at the point x = 3? Click the "Submit" button after selecting your answer.

We have established that the function is continuous at x = 3, -- but is it also differentiable at that point? To examine the derivative at this point, we must look at the derivative as we approach the point from the left and also the derivative as we approach the point from the right.

We can define the terms left-hand derivative and right-hand derivative by using the limit equation. For the derivative to exist at a point x = a, the left-hand and right-hand derivatives must be equal at that point.

Use this approach to find the value of the right-hand derivative at x = 3. Click the "Submit" button after entering your answer.

Our calculations show that the right-hand derivative equals -1, which is not equal to the value we found for left-hand derivative. Therefore, the function is not differentiable at this point.

We can conclude from this result that continuity does not imply differentiability.

Let's look at the left-hand derivative of our function. First, we set up the limit for with our function at x = 3. We then evaluate the limit and arrive at a value of 2 for the left-hand derivative.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education