# Let $f :\mathbb R\rightarrow\mathbb R$ be a function such that $f(x + 1) = f(x)$ for all x ∈ R. Which of the following statement(s) is/are true?

The given options are: (A) f is bounded. (B) f is bounded if it is continuous. (C) f is differentiable if it is continuous. (D) f is uniformly continuous if it is continuous.

Any hints on how to approach this sum? I tried using Mean Value Theorem, that did not seem to work. One observation regarding the function is that it will have same value across all integral points but how do I analyze the non-integral points?

• Note that your function is periodic with period $1$, i.e. completely described by its behavior in e.g. $[0,1]$. – Three.OneFour Feb 25 '15 at 15:24
• You need to know whether $f$ is differentiable to apply the Mean Value Theorem... – Taladris Feb 25 '15 at 16:03

You might just try to explicitly create counterexamples for some of them. You should also realize that it's easy to "periodize" a function by chopping it off at $1$ and repeating. That is, given any function $f(x)$, you can "periodize it" by making the function

$$F(x) = \begin{cases} f(x) & x \in [\,0, 1) \\ f(x-1) & x \in [\,1, 2) \\ ... \end{cases}$$

Stated slightly differently, you can precompose $f$ with the natural quotient map $\mathbb{R} \longrightarrow \mathbb{R}/\mathbb{Z}$.

Then you can take functions like $\lvert x - \tfrac{1}{2} \rvert$ and periodize it to get continuous, periodic, not-differentiable functions.

Finally, I encourage you to really try at this (and related) questions. This sort of question is very good at building intuition, which is very important.

• Is there any online resource, like video lectures, where I can learn more about such problems. I am really out of my depth here, and I am not able to find any good resource to help me understand these concepts. – Deepabali Roy Feb 25 '15 at 15:49
• No, there really aren't many video resources for this sort of question. This is a very fundamental question, both in that it requires very little domain knowledge (here, nothing besides understanding continuity, uniform continuity, and perhaps compactness) and in that it serves as a foundation for other understanding. In what context are you coming across this question now? General references include any introductory analysis book, like Spivak, Apostol or Bartle. – davidlowryduda Feb 25 '15 at 16:42
• This is a question that appeared in an entrance exam for admission into a Masters program in Maths. I've been working for the past two years and haven't been in touch with the subject. Maybe that's why I am having these difficulties. I do have an old edition of Apostol with me. Do you think I should first work through this textbook before trying out sample questions? – Deepabali Roy Feb 25 '15 at 17:18