Composition law for limits of vector valued multivariable function

Question

Let $A\subseteq\mathbb{R}^n$ and $B\subseteq\mathbb{R}^{m}$ be two non empty sets, and

$\bullet \ \ \ f:A\to B; \ \ \ g: B\to\mathbb{R^{s}}$

Suppose that $a\in\mathbb{R}^{n}$ is a limit point for $A$, $l$ a limit point for $B$, and

$\bullet \ \ \ \displaystyle\lim_{x\to a}f(x)=l\in\mathbb{R}^n \ \ \ \mbox{and} \ \ \lim_{y\to l}g(y)=m\in\mathbb{R}^{s}$

What are the hypotheses that guarantee the following equality

$$\displaystyle\lim_{x\to a}g(f(x))=m\, ?$$

Thanks.

Answer 1

I guess you are searching for minimum hypothesis. Continuity is more than enough to let this work. Can we actually easen it? Not really, it also depends on the hypotheses you have over $A$ and $B$. In general, if $f$ is not continous then such equality does not need to hold. If for instance $A$ is convex and also $f$ is convex, I think you can make it work but not even these last ones are enough.