Checking discontinuity

Question

If $f$ is a function defined on $[0,1]$ for which $ {f}'\left(\frac{1}{2}\right)$ exists but ${f}'\left(\frac{1}{2}\right) \notin [0,1]$, then $f$ is discontinuous at $x=\frac{1}{2} $.

Is this statement true and why?

Answer 1

No, the function is not discontinuous at x= ${f}'\left(\frac{1}{2}\right)$

Since the derivative of the function exists at x= ${f}'\left(\frac{1}{2}\right)$, therefore, from the definition of continuity, the function is continuous at that point.