+128475 As one of my calculus teachers would have said.....differentiability implies continuity, but continuity does not imply differentiability.....
To see why, look at the absolute value function at x = 0. Note that, it is continuous at that point but the derivative is undefined. (What is the "slope" of the function at that point .....1 or -1 ??)
We can see this in one more way......the absolute value of a number "x" is defined as √(x2).....and taking the derivative of this gives us ....... (1/2) (x2)-1/2 *(2x) = x / √(x2) = x / lxl ........and note that, this derivative is undefined at x = 0 !!!!
+4473 False. This answers your question:
http://oregonstate.edu/instruct/mth251/cq/Stage5/Lesson/diffVsCont.html
+118609 I'm just thinking.
If f(x) is continuous in an open region does that mean it could be a relation that is not a function.
Like a circle or an s shape?
Relations that are not functions are not differentiable either. 
+128475 As one of my calculus teachers would have said.....differentiability implies continuity, but continuity does not imply differentiability.....
To see why, look at the absolute value function at x = 0. Note that, it is continuous at that point but the derivative is undefined. (What is the "slope" of the function at that point .....1 or -1 ??)
We can see this in one more way......the absolute value of a number "x" is defined as √(x2).....and taking the derivative of this gives us ....... (1/2) (x2)-1/2 *(2x) = x / √(x2) = x / lxl ........and note that, this derivative is undefined at x = 0 !!!!
