Convergence of $h(f(x))$ and $h(f_n(x))$

Question

Let $h$ and $f$ be any $\mathbb{R\to R}$ functions.

Now suppose $f_n$ converges to $f$: $\mathrm{lim_{n\to\infty}}f_n=f$

Does it also mean that $h(f_n)$ and $h(f)$ are the same in the limit?

I've been browsing top questions and came across this example: Is value of $\pi = 4$?

Even though two curves become indistinguishable in the limit, the difference of their lengths is considerably different.

I think you have to at the very least, make sure that $h(f)$ is defined (or something like that; it's been years since I've thought of real analysis). — Clarinetist, Sep 27 '15 at 16:50

score 1 · Answer 1 · answered Sep 27 '15 at 16:52

1

Consider any sequence $x_n \to x$ and define $f_n(x)=x_n$. Then $h \circ f_n \to h \circ f$ if and only if $h$ is continuous at $x$. Hence the continuity of $h$ is a necessary condition. Can you decide whether it is also sufficient?

answered Sep 27 '15 at 16:52

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Convergence of $h(f(x))$ and $h(f_n(x))$

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