Here is the question:
And here is the various functions we have been looking at (I answered all the question given below):
And here is the definition of $\mathcal{F}$: $$\mathcal{F} = \{ \textbf{all continuous functions } \mathbb{R} \rightarrow \mathbb{R} \}.$$
Also, we will use the notation $p_{n}: \mathbb{R} \rightarrow \mathbb{R}$ given by $p_{n}(x) = x^n$ for $n \in \mathbb{N}$ where we take $0 \in \mathbb{N}.$ Now, since $p_{1} = \operatorname{id}_{\mathbb{R}}$, my work in problem $I$ tells us that $p_{1} \in \mathcal{F}$ and $\mathrm{const}_{y} \in \mathcal{F}$ for any $y \in \mathbb{R}$.
So, could anyone help me in the solution of problem 5? For (a), I am wondering how can I express $f + g$ and $f\cdot g$ as compositions involving $f$ and $g$ and various of other functions given in Problem I. Also, I do not know how to prove (b).
Also, if $f: \Bbb R \to \Bbb R$ and $g: \Bbb R \to \Bbb R$ are continuous then
$(f \times g): \Bbb R \times \Bbb R \to \Bbb R \times \Bbb R $ given by $(f \times g)(x,y)=(f(x), g(y))$ is also continuous.
This should be one the other functions you've looked at already.
Also defining $p: \Bbb R \times \Bbb R \to \Bbb R$ by $p(x,y)=x+y$, this $p$ is continuous as well, presumably this is well-known too.
Then $$f+g = p \circ (f \times g) \circ \Delta\tag{1}$$ The right hand side maps $x$ to $(x,x)$ first, and $f \times g$ maps that to $(f(x),g(x))$ and $p$ maps this to $f(x)+g(x)$ which equals $(f+g)(x)$ by definition.
So if $f,g \in \mathcal{F}$ (so they are continuous) then $f \times g$ also is and $f+g$ is as a composition of continuous maps under $(1)$, showing the first part of $(b)$
Multplication is the same except $p$ has to be replaced by another continuous map.