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 the two. …

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 zero as x → a …

Mar 3, 2024 · I am given this piecewise function f: R → R f: R → R, f(x) = {x2 0 x> 0 x ≤ 0 f (x) = {x 2 x> 0 0 x ≤ 0 I have to determine if the function is differentiable over R R or not. The way I think about …

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 non-vertical tangent …

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 the secant to form 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 so , …

May 16, 2018 · An example of graph which is continuous but not differentiable is given below . Can we identify the non differentiable functions from graph plot as we did earlier in continuity?

Aug 10, 2015 · I am trying to prove f is differentiable at x = 0 but not continuously differentiable there.

Jun 6, 2015 · I have a question, to aid my understanding, about the proof that differentiability implies continutity. (( Differentiability Definition When we say a function is differentiable at x0 x 0, we mean …