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

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?

## 1 Answer

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.