continuity vs differentiablity

[cole92]

Okay well I would like to try to answer the following questions on my own, but first I just need some clarification... The questions are:

True or False?
1. If a function is continuous at a point, then it is differentiable at that point.
2. If a function has derivatives from both the left and the right at a point, then it is differentiable at that point.
3. If a function is differentiable at a point, then it is continuous at that point.

What I need help on, so that I can answer these, is basically knowing what the difference between the two are. Can anyone help me with an overview of what these terms are? I understand continuity, as we have been learning about limits and what not, but I'm not too sure about differentiablity, especially in regards to continuity.

Thanks in advance to any help!

 

[galactus]

Differentiability at a point implies continuity, but the converse is not true.

A function may be continuous but not differentiable at that point.

Here is a proof.

If f is differentiable at a point, \(\displaystyle x_{0}\), then it is continuous at that point.

\(\displaystyle \lim_{h\to 0}[f(x_{0}+h)-f(x_{0})]=\lim_{h\to 0}\left[\frac{f(x_{0}+h)-f(x_{0})}{h}\cdot h\right]=\lim_{h\to 0}\left[\frac{f(x_{0}+h)-f(x_{0})}{h}\right]\cdot \lim_{h\to 0}h=f'(x_{0})\cdot 0=0\)

It follows that a function can not be differentiable at a point of discontinuity.

 

[cole92]

Differentiability at a point implies continuity, but the converse is not true.

A function may be continuous but not differentiable at that point.

Here is a proof.

If f is differentiable at a point, \(\displaystyle x_{0}\), then it is continuous at that point.

\(\displaystyle \lim_{h\to 0}[f(x_{0}+h)-f(x_{0})]=\lim_{h\to 0}\left[\frac{f(x_{0}+h)-f(x_{0})}{h}\cdot h\right]=\lim_{h\to 0}\left[\frac{f(x_{0}+h)-f(x_{0})}{h}\right]\cdot \lim_{h\to 0}h=f'(x_{0})\cdot 0=0\)

It follows that a function can not be differentiable at a point of discontinuity.

So then my answers should be:
1. False
2. True?
3. True

Is this correct?