No, a continuous function is not always differentiable. For example, f(x)=|x| is continuous but not differentiable at x=0. In this post, we will study the following: is a continuous function always differentiable?
A continuous function may not be differentiable always. Although, a differentiable function is always continuous. Lets prove this fact here.
Recall the definitions of continuous functions and differentiable functions.
Continuous Function: A function f(x) is said to be continuous at x=a if $\lim\limits_{x \to a} f(x)=f(a)$.
Differentiable Function: A function f(x) is said to be differentiable at x=a if the limit $\lim\limits_{x \to a} \dfrac{f(x)-f(a)}{x-a}$ exists.
Continuous Function is not always Differentiable
Question: Prove that continuous functions are not always differentiable.
Proof:
A continuous function may not be differentiable. For example, consider the function f(x) = |x| $=\begin{cases}x & \text{ if } x\geq 0 \\ -x & \text{ if } x< 0. \end{cases}$
Note that $\lim\limits_{x \to 0} |x|=0$ and f(0) = |0| =0.
Hence, $\lim\limits_{x \to a} f(x)=f(a)$ holds, and so the function f(x) = |x| is continuous at x=0.
On the other hand, we have:
$\lim\limits_{x \to 0+} \dfrac{f(x)-f(0)}{x-0}$ = $\lim\limits_{x \to 0+} \dfrac{x}{x}$ = 1. $\lim\limits_{x \to 0-} \dfrac{f(x)-f(0)}{x-0}$ = $\lim\limits_{x \to 0-} \dfrac{-x}{x}$ = 1. |
As the above two limits are not equal, we conclude that $\lim\limits_{x \to 0} \dfrac{f(x)-f(0)}{x-0}$ does not exist. Hence, f(x)=|x| is not differentiable at x=0.
Conclusion: The function f(x) = |x| is continuous at x=0, but not differentiable at x=0. Therefore, we conclude that a continuous function may not be differentiable.
FAQs
Q1: Are continuous functions always differentiable?
Answer: No, continuous functions are not always differentiable. For example, f(x) = |x-1| is continuous at x=1, but it is not differentiable at x=1.