Re: Are functions with null derivative locally constant?

On 24-03-2007 20:48, The World Wide Wade wrote:

Let U be a set of complex numbers and let _f_ be a differentiable
function from U into C. My question is: if f' is the null function, then
must _f_ be locally constant? By "locally constant" I mean that, for
each _z_ in U, there is some neighborhood V of _z_ in U such that the
restriction of _f_ to V is constant.

In order to eliminate as much ambiguities as possible, here is the
concept of "differentiable" that I am working with: _f_ is
differentiable if it is differentiable at each point of U and _f_ is
differentiable at a point _z_ in U if _z_ is a non-isolated point of
U and if the limit lim_{w -> z}(f(w) - f(z))/(w - z) exists.

Of course, the answer to the question that I asked above is well-known
to be affirmative if U is assumed to be an open set.

No, in fact there is a strictly increasing function on R, differentiable everywhere, whose derivative = 0 at each rational. See the discussion at

http://tinyurl.com/2d55hq

Thanks. That answers my question.

I will post a similar question that might also interest you.

Best regards,

Jose Carlos Santos
.

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