F(x) differentiable at / near x=0?

In summary, the discussion revolved around the conditions for l'Hopital's Rule to be applicable. It was determined that while f'(0) existing does not necessarily mean that f(x) is differentiable in some open interval containing x=0, it is also possible for f'(0) to exist and the limit of f'(x) as x approaches 0 to not exist. Examples of functions that meet these conditions were provided, including x^2sin(1/x^2) and a function constructed from this using the absolute value function. These examples highlight the importance of considering the overall behavior of a function, rather than just its derivative at a single point.
  • #1
Redbelly98
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This came up in a recent discussion about l'Hopital's Rule.

Suppose f(x) has a derivative at x=0, that is f'(0) exists.
Is it necessarily true that f(x) is differentiable in some open interval containing x=0?
Others--who know calculus better than I--say no, f(x) is not necessarily differentiable for x≠0. So my question is, what is an example function where that is the case? That is,
f'(x) exists at x=0
f'(x) does not exist for x close to zero
I'm unable to think of an example, but am quite curious about this.
 
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  • #2
Similar to constructing the example of a function which is continuous at only a single point, have a look at the http://www.math.tamu.edu/~tvogel/gallery/node4.html which is differentiable at only a single point. It just takes the fact that everywhere else, the tangent line has two disparate values, but at 0, all the tangent lines approach the same tangent line.
 
  • #3
Thanks! By the way, one of the other people I was discussing this with came up with this example:

f(x) = x2 sin(1/x2) for x≠0,
f(0) = 0
 
  • #4
I cannot think of one immediately as well, so probably the counterexample is pretty pathological.

Maybe you can take the f(x) from this webpage and define g(x) by
g(x) = -f(x) if x < 0
g(x) = f(x) if x > 0
g(0) = 0.

I don't really feel like proving that that's differentiable at x = 0 though :-p
It might be a good example, because around x = 0 it just looks like x |--> -x for all partial sums.

[edit]Lot of people posting while I was thinking about this :-p
Very nice example slider! And the other one from Dick RedBelly is even nicer because it is continuous everywhere :)[/edit]
 
  • #5
Redbelly98 said:
Thanks! By the way, one of the other people I was discussing this with came up with this example:

f(x) = x2 sin(1/x2) for x≠0,
f(0) = 0

Hmm, this derivative is defined everywhere, though, not just at x=0. (Before I was looking at the second derivative at x=0. My bad. :D)
 
Last edited:
  • #6
slider142 said:
Hmm, this derivative is defined everywhere, though, not just at x=0.

Aarrh, you're right. Hmmm.

The original discussion involved l'Hopital's Rule. Thinking about it some more, the actual conditions should be
f'(x) exists at x=0
lim[x→0] f'(x) does not exist​
That is, even though f'(0) exists, one cannot apply l'Hopital's Rule to problems involving f(x) that meets the above conditions.

slider, thanks for providing another example.
 

Related to F(x) differentiable at / near x=0?

What does it mean for a function to be differentiable at or near x=0?

When a function is differentiable at or near x=0, it means that the function has a well-defined tangent line at that point. This means that the function is smooth and continuous at that point, and any small changes in the input value will result in small changes in the output value.

How do you determine if a function is differentiable at x=0?

A function is differentiable at x=0 if the limit of the difference quotient (the quotient of the change in output value and the change in input value) approaches a finite value as the change in input value approaches 0. In other words, the left-hand and right-hand limits of this quotient must be equal at x=0.

Can a function be differentiable at x=0 but not at any other point?

Yes, it is possible for a function to be differentiable at x=0 but not at any other point. This means that the function has a well-defined tangent line at x=0, but not at any other point along the function's domain. This can occur if the function has a sharp point or a discontinuity at x=0.

What happens if a function is not differentiable at x=0?

If a function is not differentiable at x=0, it means that the function does not have a well-defined tangent line at that point. This can occur if the function has a sharp point, a discontinuity, or a corner at x=0. In this case, the function is said to be non-differentiable at x=0.

How is differentiability related to continuity?

A function must be continuous at x=0 in order to be differentiable at x=0. This is because a function must be smooth and unbroken in order to have a well-defined tangent line. If a function is not continuous at x=0, it cannot be differentiable at that point. However, a function can be continuous at x=0 and still not be differentiable, if the limit of the difference quotient does not approach a finite value.

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