Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined

Addition is continuous at $(a,b)$ because $|s-a|,|t-b|<\varepsilon/2$ gives $|(s+t)-(a+b)|<\varepsilon$. Multiplication is continuous there: after requiring $|t-b|<1$, one has $|st-ab|\le |s-a||t|+|a||t-b| < |s-a|(|b|+1)+|a||t-b|,$ which is less than $\varepsilon$ when both coordinate errors are smaller than $\varepsilon/(2(|a|+|b|+1))$. These coordinate conditions describe product neighbourhoods, so both operations are continuous.

The reverse triangle inequality $\big||s|-|t|\big|\le |s-t|$ makes absolute value continuous. Consequently $\max\{s,t\}=\frac{s+t+|s-t|}{2},\qquad \min\{s,t\}=\frac{s+t-|s-t|}{2}$ are continuous by step 1.1 and composition.

Reciprocal is continuous at $b\ne0$: if $|t-b|<|b|/2$, then $|t|>|b|/2$ and $\left|\frac1t-\frac1b\right| =\frac{|t-b|}{|t||b|}<\frac{2|t-b|}{|b|^2}.$ Thus division $(s,t)\mapsto s/t$ is the product of $s$ and $1/t$ and is continuous on $\mathbb R\times(\mathbb R\setminus\{0\})$; moreover $\operatorname{coz}(g)$ is open by [F1].

The map $(f,g):X\to\mathbb R^2$ is continuous by [L1], so composing it with the operations of steps 1.1 and 2.1 gives continuity of $f+g$, $fg$, $\max(f,g)$, and $\min(f,g)$; composing $f$ with absolute value gives continuity of $|f|$.

Restricting $f$ and $g$ to $\operatorname{coz}(g)$ and composing their product map with division gives continuity of $f/g$ there.

Iterating the binary operations of step 3.1 proves the finite assertions.