Limits of arbitrary functions?
November 17, 2009
Bailey Hall 201
Suppose we know the values of a function $f$ (on the real numbers, say) near but not at $x$. If $f$ is continuous, we can determine $f(x)$ by taking a limit. But what if f is an arbitrary function, not necessarily continuous? At any predetermined point $x$, we have little hope of correctly guessing $x$. Nevertheless, we can exhibit a strategy for predicting values of $f$ from nearby values that is guaranteed to be correct for almost every $x$, regardless of the function $f$; in particular, if $x$ is chosen at random in [0,1], and we are asked to guess $f(x)$ based on nearby values of $f$, the strategy will guess correctly with probability 1. While exploring how this is done, we will look at topological analogs of well-foundedness and induction.
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Created automatically on: Tue Jan 23 16:56:44 EST 2018