What is the difference between "differentiable" and "continuous" I have always treated them as the same thing. But recently, some people have told me that the two terms are different. So now I am wondering,

What is the difference between "differentiable" and "continuous"?

I just don't want to say the wrong thing. For example, I don't want to say, "$\frac{x^2}{x^4-2x^3}$ is not differentiable at $x=0$" when really, it should be "discontinuous". Please help
 A: Differentiability is a stronger condition than continuity. If $f$ is differentiable at $x=a$, then $f$ is continuous at $x=a$ as well. But the reverse need not hold.
Continuity of $f$ at $x=a$ requires only that $f(x)-f(a)$ converges to zero as $x\rightarrow a$.
For differentiability, that difference is required to converge even after being divided by $x-a$. In other words, $\dfrac{f(x)-f(a)}{x-a}$ must converge as $x\rightarrow a$.
Not that if that fraction does converge, the numerator necessarily converges to zero, implying continuity as I mentioned in the first paragraph.
A: Differentiability means that the function has a derivative at a point. 
Continuity means that the limit from both sides of a value is equal to the function's value at that point.
A: The typical example is $f(x)=|x|$.  It is continuous for all $x$, but has a corner at $x=0$ and is not differentiable there.
Your example is not defined at $x=0$.  This is stronger than not continuous, which in turn is stronger than not differentiable.
