Quote:
Silas wrote:
My favorite non-continuous function is the function: y=1 when x is rational, and y=0 when x is irrational. It's charming, since the limit at any point is both 1 and 0!
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Hmm… technically, the limit is neither 1 nor 0. The function is either 1 or 0, but it has no limit at any point.
Quote:
Richard J. Hanak wrote:
Indeed charming, but my favorite non-continuous function is the square wave, with its discontinuous amplitude. Why? Because it can (magically?) be represented by a series of continuous trigonometric functions via the Fourier expansion.
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Nothing magical about it. A series of continuous functions need not be continuous itself.