# Stone-Cech extension of a function satisfying $f(x+y)=f(x)+f(y)$?

Suppose $f:\mathbb R \to \mathbb R$ satisfies the equation $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb R$. Such a function may not be continuous, but is there still a way to extend it to a mapping from $\beta\mathbb R$ to $\beta\mathbb R$?

I am asking because I don't know much about how to extend non-continuous functions, but this function seems nice enough that maybe something will work.

• What sort of properties would you want the extension to have? – Eric Wofsey Aug 31 '16 at 7:11
• What are you asking for property of such $\beta f$ ? Obviously it cannot be continous since it restriction will not necessarly be continous. So what property do you want ? – user171326 Aug 31 '16 at 7:11
• What about the identity function or the $0$ function? Both satisfy $f(x + y)= f(x) + f(y)$. – Oles Wohnzimmer Aug 31 '16 at 7:16
• @OlesWohnzimmer But I want a general way of extending. As for properties maybe $\beta f$ can satisfy the same equation? Where $p+q=\{A+B:A\in p,B\in q\}$? – Forever Mozart Aug 31 '16 at 7:21
• @ForeverMozart Sorry..what is this notation $\beta \mathbb R$ ?Compacatification? – Dontknowanything Aug 31 '16 at 7:25
{"ft_lang_label":"__label__en","ft_lang_prob":0.9195349,"math_prob":0.99308646,"size":389,"snap":"2019-26-2019-30","text_gpt3_token_len":108,"char_repetition_ratio":0.14805195,"word_repetition_ratio":0.0,"special_character_ratio":0.2570694,"punctuation_ratio":0.083333336,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996804,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-16T00:19:31Z\",\"WARC-Record-ID\":\"<urn:uuid:ff48ff86-221e-4dd6-befc-f59b9caa881b>\",\"Content-Length\":\"140252\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9b40f30e-3363-44a4-8f9f-751a04c38001>\",\"WARC-Concurrent-To\":\"<urn:uuid:e2af5349-55b6-4cb7-8cd6-90d052c2747e>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/1909428/stone-cech-extension-of-a-function-satisfying-fxy-fxfy\",\"WARC-Payload-Digest\":\"sha1:CH6SHJ6XTNVIKUWPH3GNNLU2XA7BXA55\",\"WARC-Block-Digest\":\"sha1:US32JGWGKDXESVPEELCKEV5Q74S5PP6J\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195524290.60_warc_CC-MAIN-20190715235156-20190716021156-00169.warc.gz\"}"}