I'm currently stuck on the following problem:
Let $\phi:[a,b]\times[c,d]\to\mathbb{C}$ be a continuous function and define $g:[c,d]\to\mathbb{C}$ by $$g(t)=\int_{a}^{b}\phi(s,t)\:ds.$$ Show that $g(t)$ is continuous.
I know that to show that $g(t)$ is continuous at an arbitrary $\alpha\in[c,d]$, we must show that to each $\epsilon>0$ there corresponds a $\delta>0$ such that $$|g(t)-g(\alpha)|=\bigg|\int_{a}^{b}\phi(s,t)\:ds-\int_{a}^{b}\phi(s,\alpha)\:ds\bigg|<\epsilon$$ whenever $|t-\alpha|<\delta$. I tried to work through this, but I kept getting a $\delta$ which depended on $t$. I'm not sure where to go. Clearly, $\phi$ is uniformly continuous since it is a continuous function on a compact set. I think I need to use this uniform continuity, but I'm not sure how to incorporate this fact into the proof. Thanks in advance for any help!
Let's see.. I'm not very good at this sort of thing so tell me if I'm right: uniform continuity means that the patron on the barstool on our right in the denim jacket with the tattoo of Emmy Noether assures us that for any $\epsilon_1>0$ he can, for the right price, provide us with a $\delta_1=\delta_1\left(\epsilon_1\right)>0$ such that $\left|\phi(s,t)-\phi(s,\alpha)\right|<\epsilon_1$ whenever $\left|t-\alpha\right|<\delta_1$.
Thus, when the gentleman in the black leather jacket on the barstool to our left smashes a bottle on the bar and screams at us that there is no way that for his $\epsilon>0$ that it's possible that $\left|\int_a^b\phi(s,t)ds-\int_a^b\phi(s,\alpha)ds\right|<\epsilon$ we offer the patron on our right $\epsilon_1=\frac{\epsilon}{b-a}$ along with a packet containing a mysterious white powder and he comes back with $\delta_1\left(\frac{\epsilon}{b-a}\right)$ and we can tell the gentleman on our left with the tattoo of Kurt Friedrich Gödel $\delta=\delta_1\left(\frac{\epsilon}{b-a}\right)$ so that $$\left|\int_a^b\phi(s,t)ds-\int_a^b\phi(s,\alpha)ds\right|\le\int_a^b\left|\phi(s,t)-\phi(s,\alpha)\right|ds\lt\int_a^b\left(\frac{\epsilon}{b-a}\right)ds=\epsilon$$