In my research I came across the following question :



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> Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x\_n)\_n$, taking infinitely many values, converging to some real number $c$, such that the sequence $(f(x\_n))\_n$ converges to $f(c)$ ?
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Does every real function have this weak continuity property?
Suppose $f$ were a counterexample. Then, for any $c$, we could find a little interval $(a,b)$ containing $c$ and we could find some $\varepsilon>0$ such that all points $x\in(a,b)$ except $c$ have $|f(x)-f(c)|>\varepsilon$. (Otherwise, by taking smaller and smaller intervals and $\varepsilon$'s, we could produce a sequence converging to $c$ with $f$-image converging to $f(c)$.) By shrinking things a little, we can arrange for $a$, $b$, and $\varepsilon$ to always be rational. So uncountably many $c$'s must have the same $a$, $b$, and $\varepsilon$. Fix such $a$, $b$, and $\varepsilon$. So you've got uncountably many $c$'s, all lying in $(a,b)$ and (therefore) all having their $f(c)$'s separated by a distance of at least $\varepsilon$ --- a contradiction because there isn't that much room in $\mathbb R$.


