Give an example of a function that satisfy $\int_a^bf'(x)dx < f(b)-f(a)$ in Lebesgue's theorem

Question

According to Lebesgue's theorem on differentiation, if $f$ is an increasing function on a closed and bounded interval $[a,b]$, then $f'$ exist and is measurable and $$\int_a^bf'(x)dx \le f(b)-f(a)$$. Can someone give me an example of a function for which strict inequality occur in this case?

Answer 1

Take $f(x)$ to be a step function, say $f(x)=0$ for $x<0$ and $f(x)=1$ otherwise. On $[-1,1]$, you get $0\leq 1$.