for b
find the left and right limits
See they are equal so it is differenciable on 1
Determine whether the following two functions are differentiable at the given value
a) f(x) = xsin(1/x), f(0)=0: at given value 0
b)
f(x) = 1-x if x<1
x(1-x) if x>= 1 at given value 1
I think a is differentiable. I found the difference quotient and found that as
h->0 the limit of the quotient -> 0 also.
But I can't seem to get started on b!
Do you understand that this means you were wrong about (a) being differentiable?
The derivative at x= 0 would be given by and that limit does NOT exist.
It is also true that, while derivatives are not necessarily continuous, they do have the "intermediate value property". That means that if a function is differentiable, the limits of the two derivatives on each side must be equal.
For (b), the f'= -1 for x< 1 and f'= 1- 2x for x< 1. The limits as x goes to 1 are the same so the function is differentiable at x= 1.