Let $g,h : \Bbb R \to \Bbb R$ be any functions. Define $F: \Bbb R^{\Bbb R} \to \Bbb R^{\Bbb R}$ as $F(f)(t)=g(t)f(h(t))$. Show that $F$ is continuous.

Question

Let $g,h : \Bbb R \to \Bbb R$ be any functions. Define $F: \Bbb R^{\Bbb R} \to \Bbb R^{\Bbb R}$ as $F(f)(t)=g(t)f(h(t))$. Show that $F$ is continuous in the product topology.

Is $f$ the variable that's varying here? I am trying to express this map as components, but I don't see how.

Normally for a map $f : \prod_{i}X_i \to \prod_{i}Y_i$ I can express it as $(x_1,x_2, \dots) \mapsto (f_1(x_1), f_2(x_2), \dots)$, where $f_i: \prod_{i}X_i \to Y_i$

Here I don't have this setup. If I simply pick some $f \in \Bbb R^{\Bbb R}$ I don't have this kind of $(f_1,f_2, \dots)$ expression for it.

The idea was to show that the components would be continuous, but I don't see how to get there.

Answer 1

By the universal property of products it's sufficient to show that for every $t \in \Bbb R$ the composition $\pi_t \circ F: \Bbb R^{\Bbb R} \to \Bbb R$ is continuous. Then note that for fixed $t$

$$\pi_t \circ F = g \circ \pi_t \cdot (f\circ h \circ \pi_t)$$ which is a multiplication (in $\Bbb R$) of compositions of continuous maps, hence continuous.