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
Best Answer
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.