Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If a function series $f_n$, where each $f_n(x)$ is continuous, converges point wise to $f$ and $f$ is not continuous, does this imply that $f_n$ does not convergence uniformly to $f$.

I'm pretty sure this is true, but I didn't found a reliable source for it. Would you please help me to find such a source or provide a proof for it.

Additional, does the sentence above also hold if $f_n(x)$ is not continuous?

share|cite|improve this question
up vote 4 down vote accepted

Yes. It's a theorem that if a sequence of continuous functions converges uniformly, then the limiting function is continuous. As the contrapositive, if the limiting function is not continuous, the convergence cannot be uniform.

If the functions $f_n$ are not continuous, then they can certainly converge uniformly to a non-continuous function. (For example, all the $f_n$'s could be the same non-continuous function.)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.