# Proving using the mean value theorem

Definition: A function is said to be periodic with period $p>0$ if for every $x\in\mathbb{R}:f(x+p)=f(x)$. Prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous and $p$-periodic, then it has a global maximum and minimum in $\mathbb{R}$. That is, there exist points $x_1$ and $x_2$ such that $\forall x\in\mathbb{R}:f(x_1)\leq f(x)\leq f(x_2)$.

I think I need to use the mean value theorem but i'm not really sure. It is easy to see that there is a Min and Max point here, the proof is what makes me stuck.

• The function only needs to be continuous, not differentiable. So you cannot use the mean value theorem. Instead, look for a theorem involving continuous functions and extremal points. – Florian Dec 22 '15 at 14:46
• @Florian Extreme Value Theorem? – frosh Dec 22 '15 at 20:02

Hints : First prove that the function $f$ admits a min and a max on $[0,p]$. Secondly, try to prove that these min and max are global.
{"ft_lang_label":"__label__en","ft_lang_prob":0.85822934,"math_prob":0.99949753,"size":507,"snap":"2019-35-2019-39","text_gpt3_token_len":170,"char_repetition_ratio":0.12922466,"word_repetition_ratio":0.0,"special_character_ratio":0.3057199,"punctuation_ratio":0.1,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997352,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-19T07:11:43Z\",\"WARC-Record-ID\":\"<urn:uuid:aa526ea8-c636-4528-ab92-d9df5311b5ee>\",\"Content-Length\":\"142702\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:489cb446-6bfb-4867-8ad2-4b49214c9678>\",\"WARC-Concurrent-To\":\"<urn:uuid:82a7c1dc-40a9-4d23-8cc9-9ba818b71504>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/1585613/proving-using-the-mean-value-theorem/1585783\",\"WARC-Payload-Digest\":\"sha1:CSK4MQZLJVZYJ7AHHRAJMMZ33RMFOLX6\",\"WARC-Block-Digest\":\"sha1:YCLELWAZ5RL5VL5GTPJNARRDQDM7AQAX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573444.87_warc_CC-MAIN-20190919060532-20190919082532-00292.warc.gz\"}"}