Re: Must the graph of a derivative that exists everywhere on


On 21 Feb 2007 07:24:44 -0500, Rich Peterson <rl_pete@xxxxxxxxx>
wrote:

[ Must the graph of a derivative that exists everywhere on]
an open interval be a connected set?--The example f(x)=(x^2)sin(1/x) for
nonzero x and f(0)=0 is known to have a derivative everywhere. Although the
derivative is discontinuous at x=0, the graph is connected. I'm guessing there
is no example of the graph of the derivative being disconnected as long as it
exists everywhere on an open interval. Does anyone know? Thanks, Rich Peterson
Sacramento

Although you don't say, I imagine you're talking about real-valued
functions. The answer is no for complex-valued functions, for example
let f(0) = 0, f(x) = x^2 exp(i/x) for x <> 0.

************************

David C. Ullrich
.

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