Sorry for posting so many of these! I'm finding them a little tricky and keep running into problemsProve that is continuous at the origin.
I've done two proofs for this, one uses and , the other uses sequences.
The epsilon-delta version:
I need to find when .
Since i'm trying to find continuity at 0, let .
Therefore I need to find when .
For , . Hence pick .
For , . Hence pick
That was preliminary work so I have to write out the proper proof:
Let and define . When :
Hence f is continuous .
Now for the limits version.
Let and .
Clearly and .
I need to show that if exists.
Therefore so exists.
Hence the function is continuous at the origin.
Thanks in advance to anyone who posts!