Let $f:[0,1]\to \mathbb{R}$ be a non-decreasing function. What is the solution to the following problem $$\inf_{y\in[0,1]} \frac{1}{y}\int_0^yf(x)dx?$$
Since $f$ is non-decreasing, the average is minimized by sending $y\to 0$, in which case we get $f(0)$? Does this make sense?
It's correct except that we get $\lim\limits_{x \to 0+} f(x)$ - $f$ doesn't need to be continuous (but one-sided limit exists, as $f$ is non-decreasing).