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 $\mathbb R$.

This has already been answered satisfactorily, but let me mention that it also follows from the much stronger (and noteworthy) fact that for every $f\colon\mathbb{R}\to\mathbb{R}$ there exists a dense $D\subseteq\mathbb{R}$ such that $f|_D$ is continuous (Blumberg's theorem): see here for references. Given that this is true, take $c\in D$ and $x_n$ injective converging toward $c$ (which exists since $D$ is dense), and we have $f(x_n)$ converging to $f(c)$ by continuity of $f|_D$.

It has "been known" since 1908 that for any such function, this holds for all but countably many real numbers $c$, even when we additionally require all the sequences to approach $c$ from the same side.

In May 1908 William Henry Young presented several results for general functions from $\mathbb R$ to ${\mathbb R},$ including a result implying that, given any such function, all but countably many real numbers $c$ have the property you're asking about. These results (for more about them, see my answer here) may have been joint work with his wife, Grace Chisholm Young, and the results were published in the 1908 paper cited below.

Young showed that for co-countably many real numbers $c$ we have

$$f(c) \in C^{-}(f,c) \;\; \text{and} \;\; f(c) \in C^{+}(f,c)$$

Definition: Given a function $f: {\mathbb R} \rightarrow {\mathbb R}$ and $c \in {\mathbb R}$, we let $C^{-}(f,c)$ be the set of all extended real numbers $y$ (i.e. $y$ can be $-\infty$ or $+\infty$) for which there exists a sequence $\left\{x_{k}\right\}$ such that for each $k$ we have $x_k < c,$ and we have $x_{k} \rightarrow c$ and $f(x_k) \rightarrow y.$ In other words, $C^{-}(f,c)$ is the set of all numbers (including $-\infty$ and $+\infty$) that can be obtained as a limit of $f$-values when using some sequence converging to $c$ from the left. The right version, $C^{+}(f,c),$ is defined analogously.

Incidentally, the requirement in this definition that each $x_k < c$ (and also each $x_k > c)$ allows you to find such sequences converging to $c$ that have infinitely many values.

William Henry Young, Sulle due funzioni a più valori costituite dai limiti d'una variabile reale a destra e a sinistra di ciascun punto [On the two functions of multiple values that are determined by the left and right limits of a real variable at each point], Atti della Accademia Reale dei Lincei. Rendiconti. Classe di Scienze fisiche, Matematiche e Naturali (5) 17 #9 (1st semestre) (1908), 582-587. [Paper given at session dated 3 May 1908.]