I am slightly confused about the following example. Consider the sequence $x_n=1/n$. The $x_n$ converges to $0$. The function $f(x)=1/x$ is continuous on the interval $(0,1)$, and so it is also sequentially continuous. However, $f(x_n)=n$ clearly diverges. What am I missing here?
2026-04-06 00:43:15.1775436195
Sequentially Continuous Sequence
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Being sequentially continuous means that if $(a_n)_{n\in\mathbb N}$ is a sequence of elements of $(0,1)$ which converges to an elment $a\in(0,1)$, then $\lim_{n\to\infty}f(a_n)=f(a)$. Since $\left(\frac1n\right)_{n\in\mathbb N}$ does not converge to an element of $(0,1)$, there is no problem.