- #1

kathrynag

- 598

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g(x)=0 if x=0

Find a particular value for a such that

a) g is differentiable on R but such that g' is unbounded on [0,1].

b) g is differentiable on R with g' continuous but not differentiable at zero

c) g is differentiable on R but and g' is differentiable on R, but such that g'' is not continuous at zero.

I've only started on a) but I got a little stuck

Well I know that a function is differentiable if g' exists for all points.

I started by calculating derivatives:

g'(x)=ax^(a-1)sin(1/x)-x^(a-2)cos(1/x) if x is not zero.

Now for x=0, we have:

g'(0)=limg(x)-g(0)/x-0=limg(x)/x=limx^(a-1)sin(1/x)

I know xsin(1/x) does not exist, but for all other x, g'(0) does exist. Then g'(0) exists and equals 0 if provided a>2.

So g is differentiable provided a>2.

Now the part that confuses me is the unboundedness. I looked in this particular chapter of my book and there was no mention of boundedness at all.

So from previous chapters, I had that a sequence (xn) is bounded if there exists a number M>0 such that |xn|<=M for all n in N. I just am really confused on the unboundedness part.