View Proposal #371
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| ID | 371 |
|---|---|
| First Name | Dave |
| Last Name | Renfro |
| Institution | ACT, Inc. |
| Speaker Category | other |
| Title of Talk | The Upper and Lower Limits of a Function and Semicontinuous Functions |
| Abstract | A function is continuous on an interval exactly when the function agrees with its "limit function" on the interval, by which we mean the limit (when it exists) of the function at each point. In looking at some examples, we find that limit functions tend to be nicely behaved even when the functions are not. For example, Thomae's function is continuous on a dense set of points and discontinuous on a dense set of points, and yet its limit function is a constant function (identically equal to 0). Of course, the limit function of a function is not always defined, but by considering upper and lower limits (limsup and liminf), we get the upper and lower limit functions of a function. These also tend to be nicely behaved, as is illustrated by the characteristic function of the rationals (discontinuous at every point), whose upper and lower limit functions are constant functions. We will investigate how badly behaved the upper and lower limit functions of a function can be. This will lead to an investigation of semicontinuous functions, which are amazingly ubiquitously omnipresent throughout pure and applied mathematics. This talk should be accessible to most undergraduate math majors, although there will likely be aspects of it that are unfamiliar to nonexperts. |
| Subject area(s) | real analysis |
| Suitable for undergraduates? | Y |
| Day Preference | SaturdayStrong |
| Computer Needed? | N |
| Bringing a laptop? | N |
| Overhead Needed? | Y |
| Software requests | |
| Special Needs | I would like an elmo projector. |
| Date Submitted | 09/30/2013 |
| Year | 2013 |