What is the difference between "differentiable" and "continuous"
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 …
functions - What is the definition of differentiability? - Mathematics ...
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. …
Continuous versus differentiable - Mathematics Stack Exchange
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...
The definition of continuously differentiable functions
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 …
calculus - Is the function $x|x|$ differentiable at $0$? - Mathematics ...
May 11, 2017 · As a rule of thumb (although you should always formally verify in unusual cases), a function that is non-differentiable will really have a "kink" in the graph. If you sketch the graph of x|x| …
What does it mean for a function $f(x)$ to be differentiable at $x_0$?
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 …
calculus - Is a differentiable function always continuous ...
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 , …
real analysis - Why differentiability implies continuity, but ...
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 implies a …
Is the complex function $f(z) = Re(z)$ differentiable?
May 18, 2016 · I am reading some lecture notes online. They claim that the function f(z) = Re(z) f (z) = R e (z) is continuous but NOT differentiable. I know the definitions of a limit, of continuity, and of a …
calculus - What is the basic difference between differentiable ...
What is the basic difference between differentiable, analytic and holomorphic function? Ask Question Asked 8 years, 11 months ago Modified 3 years, 11 months ago