Continuity and Differentiability

[daytrader]

Homework Statement

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 continuous 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 continuous and differentiable. Why is it continous?

please provide some example if you can.

thanks

 

[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.

 

[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"

 

[HallsofIvy]

Homework Statement

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 continuous but not differentiable on -8.

Is that because once I take the derivative I get -8 under a root?? and why is it continous?
There is no problem with taking a third root of -8. I suspect you have copied the answer wrong. f is continuous but not differentiable on the interval [-8,8] because it is not differentiable at x= 0. It certainly is differentiable at x= -8.

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

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

please provide some example if you can.

thanks

What is your definition of sin x?