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Abstract

For suitable topological spaces $X$ and $Y$, given a continuous function $f:X\to Y$ and a point $x\in X$, one can determine the value of $f(x)$ from the values of $f$ on a deleted neighborhood of $x$ by taking the limit of $f$. If $f$ is not required to be continuous, it is impossible to determine $f(x)$ from this information (provided $|Y|\geq 2$), but as the author and Alan Taylor showed in 2009, there is nevertheless a means of guessing $f(x)$, called the $\mu $-predictor, that will be correct except on a small set; specifically, if $X$ is $T_0$, then the guesses will be correct except on a scattered set. In this paper, we show that, when $X$ is $T_0$, every predictor that performs this well is a special case of the $\mu $-predictor.