Improper integral of an increasing function

Question

If a continous function $f(x)$ is increasing and greater than zero for any $x\in[0,+\infty)$, then does the improper integral, $$\int_0 ^{+\infty}f(x) dx$$necessarily diverge?

Answer 1

Hint: For each $M\in[0,\infty)$, $\int_0^Mf(x)\,\mathrm dx\geqslant\int_0^Mf(0)\,\mathrm dx=Mf(0).$

Answer 2

Yes. Suppose that $f(c)=\delta>0$ for some $c\in[0,\infty)$. Then, for any $m\in\mathbb N$, $$ \int_0^\infty f(x)\,dx\geq\int_c^{c+m}f(x)\,dx\geq\int_c^{c+m}\delta\,dx=\delta\,m. $$ As $m$ is arbitrary, the integral is infinite.