Explore the continuity $\int_{0}^{1} \arctan{\frac{x}{y}}dx$ Explore the continuity $F(y) = \int_{0}^{1} \arctan{\frac{x}{y}}dx$ on the set $Y = {\\{y: y>0 }\\}$ I have tried to explore uniform convergence of $F(y)$ $\arctan{\frac{x}{y}} \leq \frac{\pi}{2}$ hence $F(y)$ converges by Weierstrass and hence F(y) is continuous. Am I right?

The easiest way is to do the change of variables $x=t\,y$, leading to $$ F(y)=y\int_0^{1/y}\arctan t\,dt. $$