I need help with this question. Suppose that a function f(x) is defined for all

Question

I need help with this question.

Suppose that a function f(x) is defined for all real values of x in [- 1,1]. Can anything be said about the existence of lim x rightarrow 0f(x)? Give reasons for your answer. At x = 0. lim x rightarrow 0 f(x) does not exist because it is likely that the function oscillates or jumps at that point. At x = 0. lim x rightarrow f(x) must exist because the function is defined at every point in the interval [ - 1.1]. Nothing can be said about the existence of the limit. Even though the function is defined at every point in the interval [-1,1], there may be a jump or an oscillation at x = 0.

Explanation / Answer

Nothing can be said about the existence of the limit. There can definitely be a jump at x=0.

For example lets have a continous function throughout [-1,1] say f(x) = x, but let us define f(0)=4 which make it discontinous at x=0, the limit therefore does not exist. Such discontinuities are called removable discontinuities.

Watch this video for more.

http://www.youtube.com/watch?v=f1R1u3A36T0

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