Given the sequence of functions $$ f_k(x) = \frac{1}{k(1+kx)},\qquad 0\leq x<\infty, $$ and the series $$ \sum\limits_{n=1}^\infty f_k(x). $$

I know that the series is pointwise convergent but not uniformly convergent. Following this understanding, I am asked if the series' limit function is continuous or not. The following it the limit function:

$$ f(x) = \sum\limits_{n=1}^\infty f_k(x) $$

Using Dini's theorem I know that if the following conditions apply, then we have a uniform convergence:

  • The sequence of functions is continuous and positive
  • The series converges pointwise to its limit function
  • The limit function is continuous

Since we know that there is no uniform convergence we can conclude that the limit function is not continuous.

However, on my textbook I've encountered the following proof that shows that this does indeed apply. Here's the content from the textbook. How can this be correct?

We assume there is such an x which belongs to $(0,\infty)$. We're looking for a $\delta > 0$ such that $\lvert x - x0\rvert < \delta$ implies $\lvert f(x+\delta) -f(x)\rvert < \epsilon$. Here's the proof why such a thing does exist:

enter image description here

And so we can chose such a \delta such that $$\delta < \frac{\epsilon\cdot x^2}{\sum_{n=1}^{\infty}\frac{1}{n^2}}$$

Thank you.

  • 1
    $\begingroup$ There's a hypothesis that the domain under consideration be compact in Dini's Theorem. $\endgroup$ – David Mitra Jun 27 '13 at 12:34
  • $\begingroup$ in compact you mean finite? $\endgroup$ – vondip Jun 27 '13 at 12:35
  • 2
    $\begingroup$ No, topologically compact. See this. The set $(0,\infty)$ is not compact. $\endgroup$ – David Mitra Jun 27 '13 at 12:37
  • $\begingroup$ Possible duplicate of Dini's Theorem and tests for uniform convergence $\endgroup$ – Nosrati Dec 15 '18 at 4:19

Your list of conditions that must be satisfied in order to apply Dini's Theorem to your series is not exhaustive. There is one more condition that must be met, namely that you're working over a compact set. So, you need a compact set $A$ with $\big(\sum_{k=1}^n f_k\bigr)$ converging pointwise to $f$ on $A$.

Here you have $A=(0,\infty)$, which is not compact. So, you can't appeal to Dini.

Note that your example shows that if this condition is dropped from Dini's Theorem, then its conclusion is no longer necessarily true. (In fact, none of the conditions from Dini's Theorem can be dropped. See this post for examples.)


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