- #1
ryanj123
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Homework Statement
Find sets of all x on which the following functions are continuous
using any theorems available.
When the phrase "any thms. available" is used, we are only at a stage in my beginning analysis course where we've learned up to continuity, limits, convergence/divergence, circular functions, etc. Not much beyond that, so the proof I'm trying to construct needs to fall within these limits.
Homework Equations
i) 1/(sqrt(x^2+2x+2))
ii) (x(x-1))/(x^2+2x-2)
iii) sec(x^2)
The Attempt at a Solution
So I've graphed these things to better see the continuity.
For i) obviously when the sqrt is greater than or equal to 0 is it defined.
For ii) the function is discontinuous by means of the quadratic formula at x=-sqrt(3)-1, and x=sqrt(3)-1.
For iii) this function can be represented as 1/(cos(x^2)) which is discontinuous when cos(x^2)=0.
So I understand these areas of discontinuity and continuity but I'm not sure how to formulate it exactly into an argument involving:
epsilon>0 , there exists delta>0 such that:
lf(x)-f(a)l<epsilon => lx-al<delta