Fraktal

  Let f : ℝ n → ℝ and g : ℝ n → ℝ be scalar-valued differentiable function of several ( n ) variables. Then f and g satisfy the product rule (the Leibniz rule):

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  Problem: Let be a function defined on the real line such that for every . Prove for all , i.e. is constant.   Show Solution Solution: Take any . Next, we obtain for all . So, for , . Then, take the limit as approaches , we get   Hence, the derivative is zero 0 everywhere. Therefore, must be a constant function.