Find an example of a function $f(x)$ and a convergent sequence $(x_n) \rightarrow x$ where the sequence $f(x_n)$ converges, but not to $f(x)$?

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Find an example of a function $f(x)$ and a convergent sequence $(x_n) \rightarrow x$ where the sequence $f(x_n)$ converges, but not to $f(x)$?

I thought that you would have to use a function like $f(x) = \frac {1}{x}$, but then I could not find a sequence where the convergence of $f(x_n)$ did not converge to the same value as $f(x)$.

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1 Answer

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Let $x_n=\frac{1}{n},f(x)=\begin{array}{cc} \{ & \begin{array}{cc} 0 & x\neq 0 \\ 1 & x=0 \end{array} \end{array}$

Then $f(x_n)=0,f(\lim_{n\to\infty} x_n)=f(0)=1$.

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