If a function f is differentiable at a point x = c of its domain, then

[JG89]

If a function f is differentiable at a point x = c of its domain, then must it also be differentiable in some neighborhood of x = c?

 

[Gear300]

In some neighborhood yes. Proof of derivation relies on this (if I'm correct). If 2.9999 is differentiable and 3.0000 is not, there is still the 2.99999 or 2.999999 between the 2 values, so there is some local surrounding domain you could take into account.

 

[slider142]

No this is not true. Consider the function:
[tex]f(x) = \left\{\begin{array}{lr}x^2&x\in\mathbb{Q}\\-x^2&x\not\in\mathbb{Q}\end{array}[/tex]
This function is only differentiable at the point x=0.

 

[Cantab Morgan]

Good catch, slider142! Okay, for fun let's emend the question a bit...

Given an function that is continuous in some neighborhood of c and differentiable at c, must it also be differentiable in some neighborhood of c?

 

[BobbyBear]

Ahh, good question;p

Isn't slider142's function

[tex]
f(x) = \left\{\begin{array}{lr}x^2&x\in\mathbb{Q}\\-x^2&x\not\in\mathbb{Q}\end{array}
[/tex]

continuous at x=0? Isn't it necessary for a function to be continuous at a point c for it to be derivable at c?

And what does it mean to be continuous at a point? Doesn't continuity necessarily involve a neighbourhood of a point? So I think first we must ask, if a function is continuous at a point c, then is it continuous in a neighbourhood of that point?

 

[qntty]

And what does it mean to be continuous at a point?

a function f is continuous at a if
[tex]\lim_{x \to a}f(x) = f(a)[/tex]

Continuity at a point only means that nearby points approach a, not that it's continuous near a.

 

[Preno]

Good catch, slider142! Okay, for fun let's emend the question a bit...

Given an function that is continuous in some neighborhood of c and differentiable at c, must it also be differentiable in some neighborhood of c?

I don't think so. Take any continuous function f differentiable at c and any continuous, nowhere differentiable function g bounded in some neighbourhood of 0 (such as the Weierstrass function). Then f(x) + (x-c)*g(x-c) should be continuous, differentiable at c, but not differentiable on any neighbourhood of c.

 

[BobbyBear]

Continuity at a point only means that nearby points approach a, not that it's continuous near a.

-nods at qntty-

but continuity at a point does require the existence of the function in a neighbourhood of the point. Slider's function is continuous at x=0 but at no other point (I think) :p

 

[slider142]

Indeed. The common example of a function continuous at a single point but defined for all x in R is
[tex]f(x)=\left\{\begin{array}{lr}x,&x\in\mathbb{Q}\\0,&x\not\in\mathbb{Q}\end{array}[/tex]
or you can replace the 0 function by -x, etc.
From the definition of derivative it is easy to show directly that if the function is differentiable at a point, it is also continuous there. The converse is not true, as exemplified by this function and the Weierstrauss function.