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Math Help - f(x) differentiable f'(x) not continuos?

  1. #1
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    f(x) differentiable f'(x) not continuos?

    Is there a function that is differentiable/continuous everywhere but its derivative is not?
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  2. #2
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    The function

    f(x)=\begin{cases}0\quad x<0\\<br />
x^{2}\quad x\ge 0\end{cases}

    satisfies the following conditions: f(x) is differentiable (and therefore continuous) everywhere. f'(x) is continuous but not differentiable at 0.

    Does that satisfy your requirements?
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  3. #3
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    no sorry i don't think so, i am look for a function s.t f(x) is differentiable everywhere but it's derivative f'(x) is not continuous everywhere, i think in that example f'(x) is continuous everywhere (this question came up in a tutorial today and we came to no conclusion)
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  4. #4
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    You didn't make it clear whether you needed the derivative to be just non-differentiable, or actually discontinuous (the latter being a stronger condition). I think if you have a discontinuous derivative at a point, then the anti-differentiated function (the original function you differentiated) is going to have to have some sort of corner or cusp. That would mean that the original function is not differentiable at that point.
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  5. #5
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    let f(x)=\begin{cases} -1\text{  if } x < 0 \\ 1 \text{ if } x \ge 0\end{cases}

    Now define the fucntion

    \displaystyle g(x)=\int_{-5}^{x}f(t)\, dt

    g(x) is everywhere continuous and differentiable but it has derivative f(x) is not

    This should do the trick
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  6. #6
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    The "standard example", found in many texts, is f(x)= x^2 sin(1/x) if x is not 0, f(0)= 0. That is differentiable for all x but its derivative is not continuous at x= 0.
    Note: while a derivative is not necessarily continuous, it does satisfy the "intermediate value property" so such an example is not going to be simple.
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  7. #7
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    Reply to TheEmptySet: Doesn't your function g have a corner at x = 0?
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  8. #8
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    Quote Originally Posted by TheEmptySet View Post
    let f(x)=\begin{cases} -1\text{  if } x < 0 \\ 1 \text{ if } x \ge 0\end{cases}

    Now define the fucntion

    \displaystyle g(x)=\int_{-5}^{x}f(t)\, dt

    g(x) is everywhere continuous and differentiable but it has derivative f(x) is not

    This should do the trick
    No, g is not differentiable at x= 0.
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  9. #9
    Behold, the power of SARDINES!
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    Quote Originally Posted by Ackbeet View Post
    Reply to TheEmptySet: Doesn't your function g have a corner at x = 0?
    Yes it does! So it is not differentiable there. Halls gave a better example.
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