# Continuity and Differentiability

1. Jun 18, 2009

1. The problem statement, all variables and given/known data

Can someone tell me why and why not following functions are Continous and Differentiable. I am also providing the answer but can some help me understand.. thanks

1) f(x) = x^(2/3) -1 on [-8,8]

answer: function is continous but not differentiable on -8.

Is that because once I take the derivative I get -8 under a root?? and why is it continous?

2) f(x) = SinX on [0,2pi]

answer: function continous and differentiable. Why is it continous?

plz provide some example if you can.

thanks

2. Jun 18, 2009

### Dunkle

Here's how I like to think about it: If I can draw the function with a pencil without lifting it up off the paper, then the function is continuous. However, this does not mean it is necessarily differentiable. For instance, I can draw the absolute value function without lifting my pencil up, but it is not differentiable at x=0.

3. Jun 19, 2009

### qntty

If a function is continuous but not differentiable, that means the limit $$\lim_{h \to 0}{[f(x+h)-f(x)]/h}$$ doesn't exist. Graphically, can have a sharp edge so there are infinitely many lines which can intersect the graph only at that one point (so the left hand limit doesn't equal the right hand limit), or in this case the limit of the slope diverges to infinity (positive infinity from the right and negative infinity from the left). Try graphing it to see what I mean. In addition to functions which are continuous everywhere and not differentiable at a point, there are functions where are continuous everywhere but http://en.wikipedia.org/wiki/Nowhere_differentiable" [Broken]

Last edited by a moderator: May 4, 2017
4. Jun 19, 2009

### HallsofIvy

Staff Emeritus