Question

How do you know if a function is continuous and differentiable?

Answer 1

Definition of Continuity :

A function is said to be continuous at a point $x=a$, if

$\underset{x\to \infty }{\lim f\left(x\right)}$ exists, and

$\underset{x\to \infty }{\lim f\left(x\right)}=f\left(a\right)$

It implies that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at $x=a$exists and these parameters are equal to each other, then the function $f$is said to be continuous at $x=a$.

If the function is undefined or does not exist at a point, then we say that the function is discontinuous.

Definition of Differentiability :

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

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

Hence, for a function to be differentiable at any point $x=a$ in its domain, it must be continuous at that particular point but vice-versa is not always true.