Is it possible for an integral between $a$ and $a$ to have a value other than 0 Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ with a constant $a \in \mathbb{R}$ such that
$$\int_{a}^{a} f(x) \, dx  \neq 0 \quad$$
holds?

Assuming that the integral is defined, the answer in NO.
$$\int_{a}^{a} f(x) dx =\int_{\{a\}} f(x) dx$$ and since the integral on the right is taken over the set of measure $0$, the integral is $0$.
However, notice that if instead of Lebesgue measure we put a counting measure on $\mathbb{R}$ (call it $\mu$), then
$$\int_{a}^{a} f(x) d \mu =\int_{\{a\}} f(x) d \mu=f(a)$$ Thus, any measurable function that is non-zero on $a$ will give you a non-zero integral.
