Is the Function F(x) Continuous on ℝ for Any Value of a?

[frankpupu]

Homework Statement

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)

Homework Equations

the F(x) is the differentiation of |x|^a sin(1/x)

The Attempt at a Solution

i don.t know how to consider the value a

 

[SammyS]

Homework Statement

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)

Homework Equations

the F(x) is the differentiation of |x|^a sin(1/x)

The Attempt at a Solution

i don't know how to consider the value a

Treat the variable, a, as a constant.

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

 

[frankpupu]

Treat the variable, a, as a constant.

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

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 ？

 

[SammyS]

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 ?

 

[frankpupu]

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 ?

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

 

[SammyS]

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

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,