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# Limits of arbitrary functions?

by

Chris Hardin

November 17, 2009
11:00 am
Bailey Hall 201

## Abstract:

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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Up: Research Seminars for 2009
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