# Video: Discussing the Continuity and Differentiability of a Piecewise Function at a Point

Discuss the continuity and differentiability of the function 𝑓 at 𝑥 = 0 given 𝑓(𝑥) = −9𝑥 − 6, if 𝑥 < 0 and 𝑓(𝑥) = 𝑥² − 9𝑥 − 6, if 𝑥 ≥ 0.

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### Video Transcript

Discuss the continuity and differentiability of the function 𝑓 at 𝑥 is equal to zero given 𝑓 of 𝑥 is equal to negative nine 𝑥 minus six if 𝑥 is less than zero and 𝑓 of 𝑥 is 𝑥 squared minus nine 𝑥 minus six if 𝑥 is greater than or equal to zero.

We’re given a piecewise function defined as 𝑓 of 𝑥 is negative nine 𝑥 minus six if 𝑥 is less than zero and 𝑓 of 𝑥 is 𝑥 squared minus nine 𝑥 minus six if 𝑥 is greater than or equal to zero. Let’s first sketch the functions 𝑦 is negative nine 𝑥 minus six and 𝑦 is 𝑥 squared minus nine 𝑥 minus six on the same graph to get an idea of what we’re looking at.

As we can see, the functions 𝑦 is 𝑥 squared minus nine 𝑥 minus six and 𝑦 is negative nine 𝑥 minus six both meet at the point negative six on the 𝑦-axis. So that our lower function 𝑓 of 𝑥 is defined differently either side of 𝑥 is equal to zero. As 𝑥 approaches zero, both functions tend to the same value.

We’re asked to discuss the continuity and differentiability of our function 𝑓 of 𝑥 at 𝑥 is equal to zero. So let’s define what we mean by continuity and differentiability and go from there.

A function 𝑓 of 𝑥 is continuous at 𝑥 is equal to 𝑎 (1) 𝑓 of 𝑎 exists, (2) the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 exists, and (3) the limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥 is equal to 𝑓 of 𝑎. A function 𝑓 of 𝑥 is differentiable at 𝑥 is equal to 𝑎 if 𝑓 prime at 𝑥 is equal to 𝑎 exists. For a function to be differentiable at a point 𝑥 is equal to 𝑎, a function must be continuous at that point, although the reverse is not necessarily the case.

We’re asked to discuss the continuity and differentiability of the function at 𝑥 is equal to zero. So let’s first look at the continuity at 𝑥 is equal to zero. Does our function satisfy the three criteria for continuity?

At 𝑥 is equal to zero, our function is defined as 𝑥 squared minus nine 𝑥 minus six. So at 𝑥 is equal to zero, 𝑓 of 𝑥 is 𝑓 of zero, which is zero squared minus nine times zero minus six. And that’s equal to negative six. So our first condition for continuity is satisfied. 𝑓 of zero exists, and it’s equal to negative six.

Now let’s check our second condition for continuity. Does the limit as 𝑥 tends to zero of 𝑓 of 𝑥 exist? We know that the definition of our function differs either side of 𝑥 is equal to zero. So we’re going to need to consider the behavior of 𝑓 of 𝑥 either side of zero and look at both the left-hand and the right-hand limits.

The left-hand limit, which is the limit as 𝑥 tends to zero from the negative direction, is the limit as 𝑥 tends to zero from the negative direction of negative nine 𝑥 minus six, which is negative nine times zero minus six. And that’s negative six.

The right-hand limit is the limit as 𝑥 tends to zero from the positive side, which is the limit as 𝑥 tends to zero from the positive direction of 𝑥 squared minus nine 𝑥 minus six. And that’s equal to zero squared minus nine times zero minus six, which is negative six.

Since both the left-hand and the right-hand limits exist and are equal — in fact, the function approaches negative six from both sides — our second condition for continuity at 𝑥 is equal to zero is satisfied. And since the limit as 𝑥 tends to zero of 𝑓 of 𝑥 is equal to 𝑓 of zero, which is negative six, our third condition for continuity is also satisfied. We can therefore say that 𝑓 of 𝑥 is continuous at 𝑥 is equal to zero.

So now let’s discuss the differentiability of our function at 𝑥 is equal to zero. Remember that a function 𝑓 of 𝑥 is differentiable at 𝑥 is equal to 𝑎 if 𝑓 prime of 𝑎 exists. Remember that 𝑓 prime of 𝑥 is d𝑓 by d𝑥 in Leibniz notation, which is d by d𝑥 of negative nine 𝑥 minus six for 𝑥 less than zero and d by d𝑥 of 𝑥 squared minus nine 𝑥 minus six for 𝑥 greater than or equal to zero.

Using the fact that the derivative of a constant times 𝑥 is the constant and the derivative of a constant is zero, d by d𝑥 of negative nine 𝑥 minus six is negative nine. To differentiate 𝑥 squared, we can use the power rule. This tells us that the derivative with respect to 𝑥 of 𝑎𝑥 to the power 𝑛 is equal to 𝑎𝑛𝑥 to the 𝑛 minus one. In other words, we multiply by the exponent and subtract one from the exponent.

So the derivative of 𝑥 squared with respect to 𝑥 is two 𝑥. And we already know that the derivative of negative nine 𝑥 is negative nine. So the derivative of 𝑥 squared minus nine 𝑥 is two 𝑥 minus nine. The derivative of our left-hand function is negative nine, a constant, since the function is a straight line. So as 𝑥 approaches zero from the left, the derivative is negative nine.

𝑓 prime at 𝑥 equal to zero is two times zero minus nine. That’s 𝑥 is equal to zero substituted into our right-hand function. And again, that’s negative nine. So 𝑓 prime of zero does exist and it equals negative nine. So 𝑓 prime of 𝑥 exists. Therefore, our function 𝑓 of 𝑥 is both continuous and differentiable at 𝑥 is equal to zero.