- #1
krissycokl
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Homework Statement
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.
Homework Equations
A function f from D to R is continuous at x0 in D provided that whenever {xn} is a sequence in D that converges to x0, the image sequence {f(xn)} converges to f(x0).
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 {xn} converges to x0, then there is an index N such that f(xn) = f(x0) for all indices n > N.
Thus, lim (n->∞) f(xn) = f(x0), and so f is continuous at the point x0."
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.