Find out that if f is a differentiable function at a, must f be also continuous at a or not?

A continuous function may or may not be differentiable.

Answer: If f(x) is a differentiable function, then it is a continuous function.

A function that is differentiable is always a continuous function.

Explanation: 

Given f is a differentiable function.

Consider, lim _{x → a} f(x) - f(a) = lim _{x → a} [(f(x) - f(a))/(x - a)] × x - a

= f'(a) × 0

= 0

So, this means lim _{x → a} f(x) = f(a)

Thus, if f(x) is a differentiable function, then it is a continuous function.