If f f is differentiable at x = a x = a, then f f is continuous at x = a x = a as well. But the reverse need not hold. Continuity of f f at x = a x = a requires only that f(x) − f(a) f (x) f (a) converges to …

While the conventional "existence of a limit" in 1D is nice (and as a nice side effect gives us the derivative expression too), the general definition is more powerful and it is useful to contrast …

Oct 26, 2010 · This is my personal favorite example of intuition being catastrophically incorrect in real analysis. In calculus, it is commonly taught that differentiable functions are always …

A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same c...

Jan 24, 2015 · A continuously differentiable function f(x) f (x) is a function whose derivative function f′(x) f (x) is also continuous at the point in question. In common language, you move …

Apr 28, 2017 · By definition, if x is a point in the domain of a function f, then f is said to be differentiable at x if the derivative f′ (x) exists. Intuitively, this means that the graph of f has a …

Sep 14, 2014 · Continuous Functions are not Always Differentiable. But can we safely say that if a function f(x) is differentiable within range $(a,b)$ then it is continuous in the interval $[a,b]$ . If …

Moreover, there are functions which are continuous but nowhere differentiable, such as the Weierstrass function. On the other hand, continuity follows from differentiability, so there are …

Jul 21, 2020 · Is the derivative of a differentiable function always continuous? My intuition goes like this: If we imagine derivative as function which describes slopes of (special) tangent lines …

Why are all differentiable functions continuous, but not all continuous functions differentiable? For the same reason that all cats are animals, but not all animals are cats. Differentiability also …