Multiple integrals and uniform continuity

Question

For a function $$f (x,y)$$ continuous on $[a,b]\times J$, where $J$ is an open interval:

$$g(y) = \int_a^b\ f(x,y)\ \mathrm dx$$ is also continuous.

My question is: if $J$ is a closed then $f$ is uniformly continuous, does this new quality also apply for the function $g$?

Answer 1

With $f$ uniformly continuous on $[a,b] \times J$, for any $\epsilon > 0$ there exists $\delta(\epsilon) > 0$ such that for all $x \in [a,b]$ and for all $y_1,y_2 \in J$ with $|y_1 - y_2| < \delta(\epsilon)$ we have

$$|f(x,y_1) - f(x,y_2)| < \epsilon/(b-a)$$

Notice that $\delta(\epsilon)$ does not depend upon $x$,$y_1$ and $y_2.$

Using,

$$|g(y_1) - g(y_2)| \leqslant \int_a^b |f(x,y_1) - f(x,y_2)| \, dx,$$

you should now be able to answer the question.