Continuity of a function defined by an integral

Question

Ok, Here's my question:

Let $f(x,y)$ be defined and continuous on a $\le x \le b, c \le y\le d$, and $F(x)$ be defined >by the integral $$\int_c^d f(x,y)dy.$$ Prove that $F(x)$ is continuous on $[a,b]$.

I think I want to show that since $f(x,y)$ is continuous on $[a,b]$, I can use proof by contradiction to get $F'(x)$ continuous on $[a,b]$, which would then imply that $F(x)$ is continuous. But How do I go about setting this up? Any hints would be great. Thank you in advance. Also, this is my first attempt to format everything properly, So I'm sorry if this didn't post properly.

Answer 1

Here is a start

$$ F(x+h)-F(x) = \int_c^d f(x+h,y)dy - \int_c^d f(x,y)dy $$

$$ \implies |F(x+h)-F(x)|\leq \int_c^d |f(x+h,y)- f(x,y)|dy. $$

Now, use the assumptions you have been given to finish the proof. Note that, $f$ is continuous on a compact set.

Answer 2

This is quite relevant for Rudin 10.1 from Real Analysis on page 246. Function $ f(x,y) $ is a continuous function on a compact set $[a,b]$. Therefore it is uniformly continuous, so the integrated expression is some real value, which can be made arbitrarily small.

In other words, uniform continuity of $f(x,y)$ indeed implies continuity of $F(x)$.

Great question!