Definition:Divergent Improper Integral

Definition
An improper integral of a real function $f$ is said to diverge if any of the following hold:


 * $(1): \quad f$ is continuous on $\left[{a \,.\,.\, +\infty}\right)$ and the limit $\displaystyle \lim_{b \mathop \to +\infty} \int_a^b f \left({x}\right) \ \mathrm d x$ does not exist


 * $(2): \quad f$ is continuous on $\left({-\infty \,.\,.\, b}\right]$ and the limit $\displaystyle \lim_{a \mathop \to -\infty} \int_a^b f \left({x}\right) \ \mathrm d x$ does not exist


 * $(3): \quad f$ is continuous on $\left[{a \,.\,.\, b}\right)$, has an infinite discontinuity at $b$, and the limit $\displaystyle \lim_{c \mathop \to b^-} \int_a^c f \left({x}\right) \ \mathrm dx$ does not exist


 * $(4): \quad f$ is continuous on $\left({a \,.\,.\, b}\right]$, has an infinite discontinuity at $a$, and the limit $\displaystyle \lim_{c \mathop \to a^+} \int_c^b f \left({x}\right) \ \mathrm dx$ does not exist.