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Is there an almost continuous function which cannot be almost equal to any other continuous function?

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    To any other continous function different from itself ?2012-12-07

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Consider the function $f:\Bbb R\to\Bbb R$ given by $f(x)=\begin{cases}-1 & x<0\\ 0 & x=0\\ 1 & x>0.\end{cases}$ It's continuous except at $x=0$, and differs from any continuous function on a set of positive measure.

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    I understand than$k$s.2012-12-07