$f:[a,\infty)\to\mathbb R$ be continuous such that $\int_a^\infty f(x)dx$ exists finitely , then $\lim_{x \to \infty}f(x)=0$ ?

Question

Let $f:[a,\infty)\to\mathbb R$ be a continuous function such that $\int_a^\infty f(x)dx$ exists finitely , then is it true that $\lim_{x \to \infty}f(x)$ exists and is equal to $0$ ? If not , then what if we restrict the range of $f$ to be non-negative only ?

Answer 1

No, it's not true even if you assume that $f$ is nonnegative. To see this, pick your favorite convergent series with nonnegative terms, e.g. $\sum a_n = \sum 1/n^2$. Define $f$ by placing a triangle centered at each positive integer $n$, with the triangle's area equal to $a_n$. You can do this in such a way that the heights of the triangle remain constant or even grow, by adjusting the widths appropriately (while keeping the widths less than $1$ so the triangles do not overlap).

You can replace the triangles with smooth functions with compact support to get a smooth counterexample.