How to show that the integral of the discontinuous function $\operatorname{f}\left(x,y\right) = \operatorname{sgn}\left(x - y\right)$$$\operatorname{F}\left(y\right) = \int_{0}^{1}\operatorname{sgn}\left(x - y\right)\,{\rm d}x$$
is a continuous function?
- I tried to find the integral of $\operatorname{f}\left(x,y\right) = \operatorname{sgn}\left(x - y\right)$ with respect to $x$ and as a result I get a function that depends only on $y$ and it is linear and therefore continuous for all $y$.
- But it seems to me that this is not correct in terms of the theorems and properties of integrals depending on the parameter.
- So I'm looking for a solution how to do it using the theorems from this area.