Hi I have to proove this function is discontinuous at x=0 using the formual definition of continuity:

{ 1 / squareroot(-x) x<0

f(x) = { 0 x = 0

{ 1 / squareroot(x) x > 0

Thanks a lot

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- Oct 3rd 2007, 02:49 PMtbyou87Prove Function is Discontinuous Using Sequences
Hi I have to proove this function is discontinuous at x=0 using the formual definition of continuity:

{ 1 / squareroot(-x) x<0

f(x) = { 0 x = 0

{ 1 / squareroot(x) x > 0

Thanks a lot - Oct 3rd 2007, 03:07 PMPlato
Use this sequence: $\displaystyle a_n = \frac{{\left( { - 1} \right)^n }}{{n^2 }}$.

- Oct 3rd 2007, 03:31 PMtbyou87
How did you find that sequence?

- Oct 3rd 2007, 03:52 PMPlato
Thirty-five years of teaching the stuff.

You need not understand how I found it.

**You must understand why it works as a counter-example.**

Do you see that the sequence converges to zero?

But $\displaystyle \lim _{n \to \infty } f\left( {a_n } \right) = \infty$.

What does that contradict about continuity? - Oct 3rd 2007, 05:28 PMtbyou87
Yeah that makes sense. I don't know what I was thinking. I think in my head I wanted to do a proof as you would when you are proving that a function is continuous as opposed to discontinuous. But just to clarify my last question, in general it seems that any sequence of type (1/n^p) where p>0 or a^n where |a| < 1 are likely candidates for a contradiction.

Here's my complete answer by the way:

f(x0) = f(0) = 0

(xn) = 1/n^2 subset of dom(f) = R

(xn) -> 0 = x0 (can be easily proven using defenition of a limit)

lim f(xn) = lim 1/squareroot(1/n^2) = +infinity does not equal f(x0) = f(0) = 0.

You are basically looking for a sequence that goes to 0 but f(xn) goes to something other than 0 to prove discontinuity.

Thanks that was very helpful. I've got another question this time on proving that a function is continuous. Your help was much appreciated. Thanks