Prove that f(x)={xsin(1/x), if x≠0
{ 0 , if x=0 is continuous using
i) language of limits
ii) ε - δ - language
Please help me
Thanks in advance for any replies.
Feyomi.
clearly each piece of the function is continuous when . to show that is continuous then, we need only prove that (this si by the definition of continuity at a point).
for (i) use the squeeze theorem to show this. that part shouldn't be that hard.
for (ii) apply the definition of the limit. do you know what it is?
ah yes, i understand the question much more now.
However, is it enough the show that it is continuous at x=0?
Would I not need to prove that it is continuous otherwise?
I have just applied the Sandwich Rule (as I like to call it), but I do not know how to do the second part.
I do know the definition, but I don't know how to apply it..
to prove the function is continous, we need to prove it is continuous at all points in its domain. now, for , the function is a product of two functions that are continuous when and hence the function is continuous when . the only problem is when , and so that's what we have to deal with.
as for part (ii), i will give you a hint. for
what can you do with that?