$f$ is continuously differentiable function on $\Bbb R$

Define $f_n(x)=\dfrac{f(x+1/n)-f(x)}{1/n}$

I want example of function such that $f_n\to f'$ but not uniformly on $\Bbb R$

I thought that above $f_n$ converges uniformly but this is not correct. Please can anyone help me to find example

where is my intitution going wrong?

any help will be appreciated

  • 4
    Doesn't $f(x)=x^3$ work? – yoyo Dec 7 at 13:44
  • 2
    Because that $f_{\frac{1}{n}}(x)=3x^2+\frac{1}{n^2}+\frac{3x}{n^2}$. And the term $\frac{3x}{n^2}$ is not uniformly convergent. – yoyo Dec 7 at 13:47
  • 1
    If the derivative is uniformly continuous then this convergence must be uniform. Since you specified that $f$ has a continuous derivative, the counterexample must be where that continuity is not uniform. @yoyo gives a good example. – RRL Dec 7 at 14:11
  • 1
    This is not a duplicate of the suggested target. – RRL Dec 7 at 14:12
  • @RRL Thank you~!! – yoyo Dec 7 at 14:42
up vote 2 down vote accepted

Let $f(x)=x^3$. Then it will be an example.

Since $f_{\frac{1}{n}}(x)=3x^2+\frac{1}{n^2}+\frac{3x}{n}$. But the term $\frac{3x}{n}$ is not convergent uniformly to $0$ as $n$ goes to $\infty$.

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