1. ## Functions

Let
f : R - {0} ---> Rbe defined by

f
(x) = x^asin(x^-b);

where
b > 0. Find conditions on a; b such that
(a)
f can be extended to a continuous function on R.
(b)
f can be extended to a differentiable function on R.

(c)
f can be extended to a twice differentiable function on R.

R is the set of real numbers in this case.

2. What needs to be done in part (a) is to check for which values of a and b does the function f have a limit for x tending to 0. If f does have a (finite) limit, then you can extend f to all of R by letting f(0) be that limit.

As for part b and c, you do something very similar. Well, first check if the function f is differentiable in the first place. You know that if f is to be extended to a differentiable function, then the value of f at 0 must be the limit of f(x) for x-->0. Check which choices of a and b make the extension differentiable at 0. Similarly for the second derivative.

3. Originally Posted by HappyJoe
What needs to be done in part (a) is to check for which values of a and b does the function f have a limit for x tending to 0. If f does have a (finite) limit, then you can extend f to all of R by letting f(0) be that limit.

As for part b and c, you do something very similar. Well, first check if the function f is differentiable in the first place. You know that if f is to be extended to a differentiable function, then the value of f at 0 must be the limit of f(x) for x-->0. Check which choices of a and b make the extension differentiable at 0. Similarly for the second derivative.
Forgive me for being not smart enough. Could you elaborate more? Like how to check which values of a and b will f have a limit for x tending to 0?

4. Originally Posted by ineedyourhelp

Let f : R - {0} ---> Rbe defined by

f
(x) = x^asin(x^-b);
where

b > 0. Find conditions on a; b such that
(a) f can be extended to a continuous function on R.
(b) f can be extended to a differentiable function on R.
(c) f can be extended to a twice differentiable function on R.

R is the set of real numbers in this case.

Another question from an assignment that counts towards your final grade?