Let f: [a, b] → [1, ∞) be a continuous function and let g: R → R be...

Since $f\left(x\right)$ \ge 1 \forall x \in \left[a, b\right]$for$g\left(x\right)
LHD at $x = a$ is zero
And RHD at \left(x = a\right) = \lim_{x \to {a}^{+}} \dfrac{\mathop{∫}_{a}^{x}f \left(t\right)dt - 0}{x - a} = \lim_{x \to {a}^{+}} f \left(x\right) \ge 1$<br >Hence$g\left(x\right) is not differentiable at $x = a$
Similarly LHD at $x = b$ is greater than 1
$g\left(x\right)$ is not differentiable at x = b