# Math Help - Integral and Continuity

1. ## Integral and Continuity

Question Details:
Hey guys,

I'm in a pre-analysis course and I have this homework problem over a section that my professor has neglected to cover, so I am really lost. Any help would be appreciated.
Here is the problem

Let f(x,y) be defined and continuous for $a \leq x \leq b$ and $c \leq y \leq d.$, Let

F(x)= $\int_c^d f(x,y) \, dy$
Prove that F is continuous on [a,b].

The book gave the hint "Use the uniform continuity of f." So I went through that since f is continuous and bounded, f is uniformly continuous. From there I'm not really sure where to go.

Any help would be appreciated.

Thanks

2. Hint: $|F(x)-F(x_0)|=|\int f(x,y)-f(x_0,y) dy|\leq \int |f(x,y)-f(x_0,y)| dy\leq \int \epsilon dy$.