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February 13th, 2012, 08:21 PM   #1
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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?
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February 13th, 2012, 08:29 PM   #2
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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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