Which one of the following is not true always?

• A

  If a function $f(x)$ is continuous at $x=a$, then $\lim _{x \rightarrow a} f(x)$ exists.
• B

  If $f(x)$ and $g(x)$ are differentiable at $x=a$, then $f(x)+g(x)$ is also differentiable at $x=a$.
• Correct answer
  C

  If $f(x)$ is continuous at $x=a$, then it is differentiable at $x=a$.
• D

  If $f(x)$ is not continuous at $x=a$, then it is not differentiable at $x=a$.

Hence, the correct answer is option: 'C'.