Test differentiability of the following

[tikA]

This is what I
1.For the first one I calculated the left and right hand limits.Since they werent equal they werent continuous so they arent differentiable.
2. The second one has oscillating discontinuity so they arent differentiable
3. How do I test the third one?

 

[Dr.Peterson]

View attachment 26869This is what I
1.For the first one I calculated the left and right hand limits.Since they werent equal they werent continuous so they arent differentiable.
2. The second one has oscillating discontinuity so they arent differentiable
3. How do I test the third one?

You're right about (a).

For (b), are you sure about the oscillating discontinuity? Or maybe you're talking about (c) when you say "the second one"? (Even then, it can be oscillating without having a discontinuity!)

For (b) and (c), please show how you try to check continuity, and then do the same for the derivative if applicable.

 

[lex]

For (b) and (c) just use the definition of differentiation and see does the derivative exist at 0:

[MATH]\boxed{\hspace1ex f'(0)=\lim \limits_{h\rightarrow 0} \frac{f(h)-f(0)}{h} \hspace1ex}[/MATH]
(b) [MATH]f'(0)=\lim \limits_{h\rightarrow 0} \frac{1-\cos(h)}{h^2} \hspace4ex[/MATH]?

(c) [MATH]f'(0)=\lim \limits_{h\rightarrow 0} \hspace1ex h \sin(\tfrac{1}{h})\hspace4ex[/MATH]?

and incidentally for (a) [MATH]f'(0)=\lim \limits_{h\rightarrow 0} \frac{1}{|h|} \hspace4ex[/MATH]?