# Prove using Epsilon-Delta

Let $f:I\rightarrow R$ be continuous at $y\in I$. Suppose $f(y)>m$ for some $m \in R$. Prove there exists $\delta >0$ such that $f(x)>m$ for all $x \in I$ with $|x−y|<\delta$.

Proof: Let $f:I \rightarrow R$ be continuous at $y \in I$. By definition of continuity, $\forall \varepsilon >0 \, \exists \delta > 0$ such that if $x\in I$ and $|x−y|<\delta$ then $|f(x)−f(y)|<\varepsilon$. Suppose $f(y)>m$ for some $m\in R$. And that's where I get stuck.

Let $\epsilon<f(y)-m$. Now apply the definition of continuity.
• Where do you get the $ϵ<f(y)−m$? – Maddy Oct 31 '13 at 3:18
{"ft_lang_label":"__label__en","ft_lang_prob":0.6876673,"math_prob":0.99994934,"size":451,"snap":"2019-51-2020-05","text_gpt3_token_len":170,"char_repetition_ratio":0.12304251,"word_repetition_ratio":0.0,"special_character_ratio":0.37250555,"punctuation_ratio":0.115384616,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000093,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-18T12:02:19Z\",\"WARC-Record-ID\":\"<urn:uuid:036105f5-ec90-43e0-b2c3-ec3e0aa2ebc1>\",\"Content-Length\":\"139650\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0b63e29d-f476-41cc-9c0a-fa516266c54a>\",\"WARC-Concurrent-To\":\"<urn:uuid:a85774bd-3324-4b94-9f28-2ba3303f448a>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/543656/prove-using-epsilon-delta\",\"WARC-Payload-Digest\":\"sha1:JB65TMQYHQMWWGYSPAR4DOEPH57SD755\",\"WARC-Block-Digest\":\"sha1:CD3D7HRW7Z36O7EMS2BFRP2T4NARJRYT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250592565.2_warc_CC-MAIN-20200118110141-20200118134141-00552.warc.gz\"}"}