Thread: Functions

Please help me thanks.

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.

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.

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?

Originally Posted by ineedyourhelp

Please help me thanks.

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?

Thread closed.