Continuity in definition of Induced Functional Structure

Question

I have a really simple question, however I am confused.

Bredon's Topology and Geometry gives definition of Induced Functional Structure as follows:

Suppose $F_x$ is a functional structure on space $X$ and let $f:X\to Y$ be a map. Then the induced functional structure on $Y$, where $U$ is open subset of $Y$, is given by $$ F_y(U)=\{g:U\to R|gf\text{ is in }F_x(f^{-1}(U))\} $$ I was trying to show that this is actually a functional structure and everything seems obvious (like being a subalgebra, containing constant functions etc.) except actually that $F_y(U)$ is a subset of continuous functions on $U$.

How can we show this? As far as I understand, $gf$ and $f$ are continuous but this doesn't imply that $g$ is continuous too. Or does it in this specific case?

Reference: Bredon's Topology and Geometry p. 71. I can also write definition of functional structure if it is necessary.

Answer 1

So, I came to the conclusion that we should assume that g is contionious. Otherwise, it doesn't make sense.