Proof of definite integral

Question

Let $f$ be a continuous function. Without taking an anti-derivative, prove that $$\lim_{a\rightarrow 0^{+}}\int_{0}^{a}f(t)dt=0$$

Answer 1

I believe the below approach is using mean-value theorem for integrals. An alternative is to use the fact that $f$ is continuous on a closed and bounded interval $[0,a]$. Then we have that;

$$\int_{[0,a]} f \leq \left[|\textrm{max}(f)| \cdot a \right]\to 0,\ \textrm{as}\ a \to 0$$

Answer 2

There exists $c_a\in [0,a]$ such that $\int_0^af(t)dt=af(c_a)$. Since $f$ is continuous at $0$, there exists $c>0, M>0$, such that for every $x\in [-c,c]. |f(x)|<M$. For $a\in [-c,c]$, $|\int_0^af(t)dt|\leq |af(c_a)|\leq |a|M$. This implies that $|lim_{a\rightarrow 0}\int_0^af(t)dt|\leq lim_{a\rightarrow 0}M|a|=0$.