Integration by parts appears to prove the existence of integrals that have discontinuity.
92953
Someone needs to set the record straight on what exactly antiderivatives vary by: constants functions or piecewise constant functions.
22392
How can a function with a hole (removable discontinuity) equal a function with no hole?
41043
example of discontinuous function having direction derivative
131120
Proving an integral equation.
47498
Integration and theorems on continuous functions
45602
Show that is $\int_A f$ exists, then so does $\int_A |f|$
133691
When the integral of products is the product of integrals.
30694
Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$
130586
$f$ is discontinuous $\iff$ kernel($f$) is dense in X
105767
derivative under integral intuition
142854
Convergence of Integrals - A Problem
49884
Integral vanishes on all intervals implies the function is a.e. zero
1626
How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$
86811
Help understanding a concept in multiple integrals
128171
Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?
112675
Integration by parts of $\cot x$
50887
Prove continuity/discontinuity of the Popcorn Function (Thomae's Function).