Definition:Divergent Improper Integral

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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 $\hointr a \to$ 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.


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