Category:Divergent Improper Integrals

This category contains results about Divergent Improper Integrals.
Definitions specific to this category can be found in Definitions/Divergent Improper Integrals.

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

$(1): \quad f$ is continuous on $\hointr a \to$ and the limit $\ds \lim_{b \mathop \to +\infty} \int_a^b \map f x \rd x$ does not exist

$(2): \quad f$ is continuous on $\hointl \gets b$ and the limit $\ds \lim_{a \mathop \to -\infty} \int_a^b \map f x \rd x$ does not exist

$(3): \quad f$ is continuous on $\hointr a b$, has an infinite discontinuity at $b$, and the limit $\ds \lim_{c \mathop \to b^-} \int_a^c \map f x \rd x$ does not exist

$(4): \quad f$ is continuous on $\hointl a b$, has an infinite discontinuity at $a$, and the limit $\ds \lim_{c \mathop \to a^+} \int_c^b \map f x \rd x$ does not exist.