Question

What are differentiability and continuity?

Answer 1

Step-1: Continuity:

A function $f\left(x\right)$ is said to be continuous at a point $x=a$ if,

• $f\left(a\right)$ is defined
• $\underset{x\to a}{\lim }f\left(x\right)$ exists
• $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$

Step-2: Differentiability:

A function $f\left(x\right)$ is said to be differentiable at the point $x=a$, if the derivative $f\text{'}\left(a\right)$ exists at every point in its domain.

$f\text{'}\left(a\right)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$

For a function $f\left(x\right)$ to be differentiable at any point $x=a$ in its domain, it must be continuous at that particular point but a continuous function need not to be a differentiable function.