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February 13th, 2012, 08:21 PM  #1 
Newbie Joined: Jan 2012 Posts: 29 Thanks: 0  Help for Finding functions for certain conditions
For each of the following conditions, define a function which satisfies the condition. a) A function which is defined at all real numbers, but is discontinuous at x = 3. b) A function which has a vertical asymptote at x = 2, but is continous at all other real numbers. c) A function which is discontinuous at infinitely many points. (side question) If f(x) < g(x) < h(x) for all X E R, and if f and h are continuous functions, must g also be continuous? If so why? If not, can you come up with a counter example? Totally lost right now, also struggling with limits at the moment so can someone help me with these? 
February 13th, 2012, 08:29 PM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Help for Finding functions for certain conditions
a) http://en.wikipedia.org/wiki/Classifica ... ntinuities ... see the examples. Either of the first two would suffice, with appropriate selection of x_0, of course! b) Make it (some polynomial without x  2 as a factor) over (x  2). Such a rational function should do the trick. c) Step function Extra Just think of something SUPER simple, like f and h being horizontal lines. So let's say you've got two horizontal lines. Must any function that lies entirely between them be continuous? No way! 

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