If one uses the limit definition of a derivative (lim of (f(x)-f(a)) / (x-a)) as x approaches a) on a function and you get a value (ie. it is not undefined) does that mean the derivative of the function at that point exists? In other words, even if the limit definition of the derivative works, do you still need to determine whether the function is continuous, smooth and non-vertical at x=a in order to know that the function is differentiable at x=a? For example, The derivative of f(x)=(x^2)(sinx) at x=0 is 0 (using limit definition). Is that all the proof needed to show that the function is differentiable at x=0?