Question

If$\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$ exists, what can you say about the differentiability of $f$at $x=c$? What can you say about the continuity of $f$at $x=c$?

Short answer

The function is differentiable as well as continuous at$x=c$.

Step-by-step solution

Step 1. Given information

It is given that $\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$exists, we have to say about the differentiability and continuity of$f$at$x=c$.

Step 2. Check differentiability and continuity

Since $\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$is exist, therefore

$\underset{x\to c^{-}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}=\underset{x\to c^{+}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$

Thus the function is differentiable at $x=c$.

The function is differentiable at $x=c$only when it is continuous at $x=c$.

Hence the function is continuous at$x=c$.