Let $X$ be a topological space, and $f,g: X \to \mathbb{R}$ be continuous functions. Then show (1) and (2).

Question

Below is the way I did it.

Please verify my proof, let me know if there is any concern or questions. Also, help me on notation and style. Appreciate your help & support.

Answer 1

A major flaw is that you assume compactness or local compactness to find maxima or minima of $f$ or $g$ on certain open sets. You cannot do that, and you don't need it either:

Just use as a lemma that $p: \Bbb R \times \Bbb R \to \Bbb R$ defined by $p(x,y)=x+y$ is continuous. You can either prove that similarly as you did or know that it holds from an earlier course in real analysis.

Then given $f,g:X \to \Bbb R$ the map $f \nabla g: X \to \Bbb R \times \Bbb R$ defined by $f \nabla g(x)=(f(x), g(x))$ is continuous, because $\pi_1 \circ f \nabla g = f$ and $\pi_2 \circ f \nabla g = g$, where $\pi_1,\pi_2$ are the two projections on $\Bbb R^2$. By the universal property for continuity of maps into products, we then have immediately that $f \nabla g$ is continuous, and $f+g = p \circ (f \nabla g)$ is then continuous, as a composition of continuous maps.

Composing with $\mu(x,y)=xy$ or $m(x,y)=x-y$ (also $\Bbb R \times \Bbb R \to \Bbb R$) gives $fg$ and $f-g$ instead.

$m$ is continuous at $(p,q)$ because if $(x,y) \in \Bbb R \times \Bbb R$, then given $\varepsilon>0$:

$$|m(p,q)- m(x, y) = |pq - xy| = |pq - py + py -xy| \le |p||q-y| + |y||x-p|\tag{1}$$

so we only have to take $(x,y)$ so that $|q-y| < \frac{\varepsilon}{2|p|}$ to ensure the first term of $(1)$ will be $< \frac{\varepsilon}{2}$, and knowing that, $|y| \le |q|+ \frac{\varepsilon}{2|p|} = \frac{2|p|+ \varepsilon}{2|p||q|}$ (fixed for given $\varepsilon, |p|,|q|$!) and that gives us a bound for $|x-p|$ so that the second term of $(1)$ (at the end) is $< \frac{\varepsilon}{2}$ as well.

For problem 2, we need that $g$ doesn't map to $0$ for the points in the open domain, so the open set on wgucg $\frac{f}{g}$ is defined, is probably a subset of $g^{-1}[\Bbb R\setminus \{0\}]$ which is open by continuity of $g$. Then we need a computation as you did for $|\frac{p}{q}-\frac{x}{y}|$ to find similar estimates for continuity at $(p,q)$ of the function $\delta(x,y):=\frac{x}{y}$ to compose with $f \nabla g$ again. Etc.

So some work left to do for you: adapt your proof to show continuity (at a point, not uniformly, as you tried!) of $\delta$ at $(p,q)$, where $q \neq 0$. That reduces the problem to a single function on $\Bbb R^2$ and the $f,g$ are taken care of by composing, as we saw.