# A continuous function problem

1. Feb 14, 2012

### frankpupu

1. The problem statement, all variables and given/known data
assume a function F(x)=(a|x|^(a-1))*(sin(1/x))-((|x|^a)/(x^2))*(cos(1/x)) for x not equal to 0
F(x)=0 for x equal to 0

for what values of a that this function is continuous on R(real number)

2. Relevant equations
the F(x) is the differentiation of |x|^a sin(1/x)

3. The attempt at a solution
i don.t know how to consider the value a

2. Feb 14, 2012

### SammyS

Staff Emeritus
Treat the variable, a, as a constant.

What must be true in order for F(x) to be continuous at x?

3. Feb 14, 2012

### frankpupu

i am consider that if both the two parts of the function can be differentiable then both of then are continuous,then done. but how i know if a>3 then they are both differentiable at 0, but how about the other points. does this method make sense ？

4. Feb 14, 2012

### SammyS

Staff Emeritus
OK: So we have
$F(x)=\left\{ \begin{array}{cc} 0\,,&\text{ if }x=0 \\ a|x|^{a-1}\sin(1/x)-(|x|^a/x^2)\cos(1/x)\,,&\text{ otherwise} \end{array} \right.$​
All the functions of which F(x) is composed are continuous for all x except some are not continuous at x = 0.

So It appears that you need to see what values of the variable, a, makes F(x) continuous at x=0.

What's the test to see if F(x) is continuous at x=0 ?

5. Feb 15, 2012

### frankpupu

that means lim x->0 F(x) exists,right? then i can prove it

6. Feb 15, 2012

### SammyS

Staff Emeritus
Nothing I wrote shows that F(x) is continuous at x=0 !

Does $\displaystyle \lim_{x\to0}\,F(x)$ exist?

If so, is $\displaystyle \lim_{x\to0}\,F(x)=F(0)\,,\ \text{ which is }0\,?$

To answer yes to these questions may impose restrictions on the value of the variable, a,