Finding $f(x)$ such that $\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$

Question

Does there exist any method to find the function $f(x)$ which satisfies $$\int_{a}^{b}f(x)dx=\sum_{k=a}^{b}f(k)$$

For example $$\int_{- \infty}^{\infty}\frac{\sin(x)}{x}dx=\sum_{k=-\infty}^{\infty}\frac{\sin(k)}{k}=\pi$$

Answer 1

Unless we allow Kroneker delta 'singularity' functions, it's easy to see $f(k)=0$ for all integer $k$ since $f(a) = \sum_{k=a}^af(k)=\int_a^a f(x)dx = 0$.

Hence with $a, b, k$ integers, only such functions are those whose integrals vanish over every integer range and is $0$ at all integers.

An example: $$f(x) = \left\{\begin{array}{l}0&\text{if }x\in\mathbb{Z}\\sin(2\pi x)&\text{if }\lfloor x\rfloor\text{ is even}\\2(x-[x])-1&\text{if }\lfloor x\rfloor\text{ is odd}\end{array}\right.$$

You can get creative.