1. ## Discontinuous function

Hi,

I have a piece-wise function similar to this:

f(x) = (-ln x)/3 (x>0)
-2/x (x<0)

The limit of this is infinity, it exists as the one sided limits exist for f(x) and both equal infinity (although can it be argued that lim x->0 -2/x DNE?).

So how does one classify this discontinuity, is it a removable or essential discontinuity? Why?

Any help is appreciated.

Cheers

2. ## Re: Discontinuous function

Originally Posted by iMagoo
Hi,

I have a piece-wise function similar to this:

Code:
f(x) = (-ln x)/3 (x>0)
-2/x (x<0)
The limit of this is infinity, it exists as the one sided limits exist for f(x) and both equal infinity (although can it be argued that lim x->0 -2/x DNE?).
So how does one classify this discontinuity, is it a removable or essential discontinuity? Why?
What definition of a removable and/or essential discontinuity do your notes/textbook use?

In most cases it is a removable only if it is possible to redefine the function so as to have a finite limit.

3. ## Re: Discontinuous function

If lim(x->a) f(x) DNE, essential discontinuity at a.
If lim(x->a) f(x) = L =/= f(a) then it's removable.

Turns out, underneath the limit laws, my book also defines: lim (x->a) f(x)/g(x) = L/M if M=/=0, DNE if M=0, L=/=0.

Using this definition, lim (x->0) -2/x DNE, therefore two sided limits are not equal, therefore the discontinuity is essential. Is this correct?