Proving Discontinuity of f(x) = 1/(x^2)

[ccsun]

Homework Statement

Prove that f(x) = 1/(x^2) is not continuous at x = 0 using the epsilon and delta definition of a limit

Homework Equations

definition of discontinuity
There exist epsilon > 0 such that for all delta > 0 there is an x such that |x-0| < delta but |1/(x^2)| >= Epsilon

The Attempt at a Solution

I don't know how to go from here. Like how do i prove it? I don't understand!

 

[Coto]

Start by assuming that $|x| < \delta$. Consider how this piece of information affects your function $1/x^2$. Can you derive an $\epsilon$ that is a function of delta such that $|1/x^2| \geq \epsilon$?

 

[Deveno]

now, if 1/x² IS continuous at 0,

$$\lim_{x \to 0} f(x) = f(0)$$ which is a problem, because f(0) doesn't exist.

but let's go even further, let's prove that there isn't ANY real number we can pick for f(0) that will make 1/x² continuous at 0.

now, for this limit L to exist, we need to be able to find a δ > 0, so that:

0 < |x| < δ implies |1/x² - L| < ε.

can L be 0? choose ε = 1. that would mean that we could find δ > 0 so that 0 < |x| < δ implies |1/x²| < 1. here, we have another problem.

|1/x²| = (1/|x|)², so if 0 < |x| < δ, (1/|x|)² > δ². so no δ less than 1 will work. but if we choose δ ≥ 1, then (0,1) is a subinterval of (0,δ), and for 0 < |x| < 1, |1/x²| > 1.

since for the particular ε = 1, we can't find a δ, L can't be 0. so L > 0 (since 1/x²) (if L < 0, then on the interval (0,1], 1/x² would have to take on the value 0 (by the intermediate value theorem) but if x ≠ 0, 1/x² > 0).

so what happens if δ < 1/(√(2L))?

then 0 < |x| < 1/(√(2L)) means that |1/x² - L| ≥ |1/x²| - |L|

= 1/|x|² - L > 1/(1/(√(2L))²) - L = 2L - L = L

now if we choose 0 < ε < L (which we can, since L > 0, for example ε = L/2 would work fine), we have the same problem as before, no δ < 1/(√(2L)) will work, and any larger choice for δ will lead to |x| < 1/(√(2L)) for some x as well (larger δ's don't help).

so there is always some ε > 0 that doesn't have a δ, no matter what we choose for L. so such an L doesn't exist, we cannot define f(0) to be any real number in such a way as to make f(x) = 1/x² continuous.

 

[Eva63]

Hello - where did the 1/sqrt(2L) come from? thank you.