Does $f$ have to be a continuous function in the definition for an improper integral of an unbounded function

Question

I recently learned the following definition for type 2 improper integrals (unbounded functions):

Let $a < b$.

Let $f$ be a continuous function on $(a, b]$.

We define the integral of $f$ from $a$ to $b$ as

$$ \int_a^b f(x) \, dx = \lim_{c \to a^+}\left[ \int_c^b f(x) \, dx \right] $$

assuming this limit exists.

The integral is convergent when the limit exists.

The integral is divergent when the limit doesn't exist.

Must $f$ be a continuous function in this definition?

Would it be enough, or equivalent, to claim that $f$ is defined on $(a, b]$ and assume that the integral exists?

Answer 1

The function doesn't have to be continuous. On the other hand, it must be assumed that the restriction of $f$ to each interval $[c,b]$, with $c\in(a,b)$, is integrable.