abstract math problem..?

how can I prove " f is differentiable then f' has intermediate value property" ?

Comments

  • Answer #2 is not a good counterexample. That function doesn't have a derivative at x=0: f'(0)=lim{[f(0+h) -f(0)]/h} = lim f(h)/h = 0 if h<0 and =1 if h>0 so the limit is undefined. So f is not differentiable at x=0.

    The wikepdia site (way at the end) says it is true. I couldn't find any proofs online too easily. The classic differentiable but with a discontinuous derivative example is x^2*sin(1/x) but (I guess) that function does satisfy the intermediate value property, even though it is discontinuous.

    Here ya go! It is called Darboux's theorem and the proof is:

    Without loss of generality we might and shall assume f' + (a) > t > f' − (b). Let g(x) := f(x) - tx. Then g'(x) = f'(x) − t, g' + (a) > 0 > g' − (b), and we wish to find a zero of g'.

    Since g is a continuous function on [a,b], by the extreme value theorem it attains a maximum on [a,b]. This maximum cannot be at a, since g' + (a) > 0 so g is locally increasing at a. Similarly, g' − (b) < 0, so g is locally decreasing at b and cannot have a maximum at b. So the maximum is attained at some c in (a,b). But then g'(c) = 0 by Fermat's theorem (stationary points).

    found at http://en.wikipedia.org/wiki/Darboux%27s_theorem_%...

  • Not enough information.

    In addition to being differntiable on an open interval, f must also be continuous on a closed interval. Otherwise you can find counterexamples such as the one above.

    Given continity and differntiability, this follows from the completeness property of real numbers.

    Complex numbers have a similar completeness property.

    http://en.wikipedia.org/wiki/Complete_space

  • I think this is false.

    Look at f(x) = 0 if x <=0,

    f(x) = x >= 0.

    The derivative is the unit step function, which

    is not continuous at 0 and there is no value

    of x for which f'(x) = 0.

    Hope this makes sense!

  • Do you really need to prove it, or just understand it? In either case, information and the proof can be found on the Wikipedia page.

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