Let $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$ be continuous. Prove continuity of the function $F:[a,b]\rightarrow\mathbb{R}$ defined by
$F(s)=\int_c^d f(s,t)dt$
*Here's what I tried to do (with questions):
Since $f$ is continuous, $f$ is Riemann integrable. (Can I conclude $f$ is separately continuous? that is $f(s,t)$ is continuous in $s$ for each fixed $t$.)
Since $f$ is integrable, so linearity holds (If not integrable, then linearity cannot hold, right?) Thus, $|\int_c^df(x,t)dt-\int_c^df(y,t)dt|=|\int_c^d(f(x,t)-f(y,t))dt|$. ($t$ is the variable, and $x,y$ are fixed. So if $f(s,t)$ is continuous in $t$ for each fixed $s$, can we also conclude the same equality?)
Since $f$ is uniformly continuous, because the domain is compact, $\exists\delta>0$, such that $|x-y|<\delta$, then $|f(x,t)-f(y,t)|<\frac{\epsilon}{|c-d|}$ (Would $f$ be separately continuous be enough for this inequality?)
If the above is correct, then the result follows, i.e.$F$ is continuous.
Your argument is fine.
To prove the continuity of $F$, separately continuous would be enough if $f$ is bounded by an integrable function.