(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Define

f = { [itex]x^2[/itex] if [itex]x \geq 0[/itex]

[itex]x[/itex] if [itex]x < 0[/itex]

At what points is the function f | [itex]\Re -> \Re[/itex] continous? Justify your answer.

2. Relevant equations

A function f from D to R is continuous at x_{0}in D provided that whenever {x_{n}} is a sequence in D that converges to x_{0}, the image sequence {f(x_{n})} converges to f(x_{0}).

3. The attempt at a solution

Check x=0. {1/n} converges to 0. {f(1/n)} converges to 1. f(0) = 0. 0 ≠ 1. Thus the function is not continuous at x=0.

When it comes to proving it's continuous elsewhere...that's where I have a problem.

The example in the book for proving that a function is continuous at every other point simply states that "If a sequence {x_{n}} converges to x_{0}, then there is an index N such that f(x_{n}) = f(x_{0}) for all indices n > N.

Thus, lim (n->∞) f(x_{n}) = f(x_{0}), and so f is continuous at the point x_{0}."

How is that a proof? That logic doesn't even exclude x=0. Can someone either elucidate what they've done, or explain an alternate way of proving this function's continuity at all points other than x=0?

Thanks in advance.

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# Homework Help: Proving Continuity Everywhere

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