TITLE: Exists a function f continuous at a point x which is a limit point of points at which f is not continuous?
QUESTION [0 upvotes]: Problem: Is there a function f defined on all real numbers that is continuous at x and not continuous at a set of points in the domain having x as a limit point? 
I have determined that we want f at x. We want p(n)-> x, but we do not want f continuous at p(n). Also, any domain has domain of all real numbers. 
After countless practice trying to interpret this question, I am unsure if it is true or false. 
I believe this definition is crucial with this problem: We say that a function f is continuous at the point x if and only if for every sequence p1, p2, p3,... in the domain of f converging to x we have that f(p1), f(p2), f(p3),... converges to f(x).
Any help on how to get started would be greatly appreciated.

REPLY [3 votes]: Let $f(x)=x \chi_{\mathbb Q}(x)$, that is, $f(x) = x $ if $x$ is rational and $0$ otherwise.  It is continuous only at $0$.
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