if $f$ is integrable on $[a,b]$ , show that $\lim_{s \to a^+} \int _{s}^{b}f=\int _{a}^{b}f$

Question

If $f$ is integrable on $[a,b]$ , show that $\lim_{s \to a^+} \int _{s}^{b}f=\int _{a}^{b}f$

I proved that if $f$ is integrable on $[s,b]$ then $f$ is integrable on $[a,b]$

But how to prove the limit ?

Answer 1

You seem to be using the Riemann integral, according to the tags you are using. So I assume that $f$ being integrable means, among other properties, that $f$ is bounded. In this case the reasoning is rather easy. Since $\int_a^b f -\int_s^b f= \int_a^s f $ you can bound the right hand side of this equation easily (using step functions) by $ (s-a)\max{|f| }$ which tends to $0$ since the $\max$ is bounded.