I'm used to using the defintion to prove continuity, not discontinuity.
But if f is not defined at x=0, what is there to prove? A function cannot be continuous at a point where the function is undefined, right?
You can use sequences to come up with an epsilon?But we can probably use Plato's sequence to come up with an appropriate .
I can already prove it is discontinuous using sequences, I use and . I tried proving discontinuity using and I kept getting stuck!
This is a question one of my friends posed me so i'll change the initial question so it's defined at 0.
I just had another go:
I need to find s.t .
Using :
set
Therefore since
Hence as required.
Is this right?
I think I know what you mean.
If we pick an between we can use it to reach a contradiction in the definition of discontinuity.
For example, I could choose . If this is the case, then from my previous post, . Wouldn't I need to find a more general expression for delta though?
Showcase to prove discontinuity at x=0 you must show:
After you have chosen ε>0 you must prove:
for all δ:
if δ>0 ,then there exists an ,x such that :
|x|<δ AND |f(x)-f(0)|
AND in symbols:
and ,such that:
|x|<δ AND |f(x)-f(0)| .
Notice the difference with your definition.
Do you agree??