Can the definite integral be written as the difference of two integrals with lower limits of $-\infty$

Question

My question is: Can $\int_a^b f(x)dx$ be written as

$$\int_a^b f(x)\ dx = \int_{-\infty}^b f(x)dx - \int_{-\infty}^a f(x)dx$$ or is a limiting process required ??

Answer 1

If all 3 integrals exist, the equality will hold. But it is very much possible for the definite integral to exist and the other two not.