Show that the integral of a positive function is positive
Say $f$ is continuous at $x$. Then there exist $r>0$ and $\delta>0$ so that $f\ge r$ on $[x-\delta,x+\delta]$. This shows easily that $$\int_0^1 f\ge\int_{x-\delta}^{x+\delta}f\ge2\delta r>0.$$