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Is such a theorem exists? (Uniform convergence).

  1. Sep 25, 2009 #1

    MathematicalPhysicist

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    I would like to know if there's a counterpart to the single variable theorem, that if f is a differentialble function with a bounded derivative, is uniformly continuous.

    I think the counterpart should be, if f(x1,...,xn) is continuous function, and differentiable, and each f'_xi are bounded then f is uniformly continuous.

    But I have my suspicions.

    Anyone can corroborate or disprove this?
     
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  3. Sep 25, 2009 #2

    quasar987

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    I think f_n:[0,1]-->R defined by f_n(x)=x^n is a counter-example.
     
  4. Sep 25, 2009 #3

    Office_Shredder

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    I'm not sure what a sequence of functions has to do with anything here....

    If each of the partial derivatives are bounded, the total derivative (which is the matrix of every partial derivative) has each entry bounded. The directional derivative, which tells you how fast the function changes when you go in a certain direction, is equal to the total derivative matrix multiplied by a (unit) vector in the direction you want to look at. So this is bounded also, which means that the function cannot increase faster than a certain value (if the partial derivatives are all bounded by M, then this is M*n where n is the dimension of your domain. If your image space is more than one dimension, say dimension k, it's M*n*k)
     
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