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A function bounded and differentiable, but have an unbounded derivative?

  1. Jul 20, 2011 #1
    Can a function f: (a,b) in R be bounded and diffferentiablle, but have an unbounded derivative. I believe it can, but can not think of any examples where this is true. Anyone have any ideas?
     
  2. jcsd
  3. Jul 20, 2011 #2
    Sure, so long as you properly choose your domain.

    So let's think about this, we want a function whose derivative approaches infinity, but such that the function itself is bounded. Probably the easiest way to do this is to find a function whose derivative goes to zero, then rotate about the axis y=x.

    You don't need anything complicated. A monomial will do fine.
     
  4. Jul 20, 2011 #3

    Fredrik

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    Try to think of a function with a graph that looks a bit like the graph of a periodic function, but is such that the distance between the peaks get shorter and shorter instead of staying the same.
     
  5. Jul 20, 2011 #4
    Very nice. Perhaps somewhat more complicated, but nice.
     
  6. Jul 20, 2011 #5

    Ray Vickson

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    Sqrt(x), 0 < x < 1.
     
  7. Jul 20, 2011 #6

    Fredrik

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    Ray, here in the homework forum it's important to answer with hints, not complete solutions.
     
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