# Why is O notation the worst case? [duplicate]

I don't understand why O notation is the worst case. If this notations describes a function f such that 0 <= f(n) <= cg(n), we can see that in any case f will be smaller that the original function g that describes the running time. In my idea, f(n) describes a better option because the running time is less for any input n. Someone can explain me please why O notation is the worst case?﻿

## marked as duplicate by David Richerby, Yuval Filmus algorithms StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Feb 5 at 2:35

It doesn't, if $$f \in O(g)$$, it "just" tells us that $$g$$ is an asypmtotic upper bound for $$f$$.
{"ft_lang_label":"__label__en","ft_lang_prob":0.9096076,"math_prob":0.9197983,"size":430,"snap":"2019-43-2019-47","text_gpt3_token_len":97,"char_repetition_ratio":0.14319248,"word_repetition_ratio":0.051282052,"special_character_ratio":0.23255815,"punctuation_ratio":0.07777778,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97032017,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-22T05:26:43Z\",\"WARC-Record-ID\":\"<urn:uuid:a562ebd2-0e4b-4d75-8f8e-4e0f9c882b04>\",\"Content-Length\":\"116574\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ac2ca7d-3344-4fa9-9154-fab843f92140>\",\"WARC-Concurrent-To\":\"<urn:uuid:ddc26d40-3481-40d0-94da-79a27a0f8e98>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://cs.stackexchange.com/questions/103851/why-is-o-notation-the-worst-case\",\"WARC-Payload-Digest\":\"sha1:T3C3SQNCXUJZ2H6FYA7C35HFKJWTAAGJ\",\"WARC-Block-Digest\":\"sha1:4RT7RLMQH24F6UHLQNFO75OK7OO3IR7F\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496671239.99_warc_CC-MAIN-20191122042047-20191122070047-00203.warc.gz\"}"}