Consider a real valued function $f$ with $f(0)=0$ and $f(1/n)=1$ which is linear on $[0,1/n]$ and $0$ everywhere else. Does $f$ converge pointwise to $0$?

Specifically im trying to understand the behavior at $0$. The value at $0$ as $n$ goes to infinity seems illdefined.

  • $\begingroup$ Yes, $f$ converges pointwise to $0$. The sequence of values at $0$ is, by definition, the constant sequence with constant value $0$, which converges to $0$; what is the problem? $\endgroup$ Aug 5 at 22:00
  • $\begingroup$ The limit of the location of the peak is 0. So while $f(0)=0$ for all $n$, i expect the limit is undefined at this point. $\endgroup$
    – dmh
    Aug 5 at 22:38
  • 1
    $\begingroup$ $f_n(x)$ is pointwise convergent to $f(x)$ if $$lim_{n\to\infty}f_n(x)=f(x)$$ The convergence of $f_n(0)$ is independent of the values of $f_n(x)$ where $x \neq 0$ $\endgroup$ Aug 6 at 0:44


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