What happens with a differentiable function in the neighborhood of x?

In summary, the conversation discussed a problem in mathematical analysis regarding the direction field of a differential equation. It was questioned whether a function with certain properties would always have a certain property, and if not, what additional assumptions would make it true. It was concluded that the statement is not always true, with a counterexample provided, and that infinite differentiability and a finite number of sign changes in the derivative would be sufficient to make the statement hold. The use of the mean value theorem was also discussed.
  • #1
Skalman
3
0
A problem in mathematical analysis that I have problems getting to grips with (need it to characterize the direction field of a differential equation)

Suppose f(x) is a continuously differentiable function on ℝ with f(0)=f'(0)=0. Suppose that for any ε>0 there is some x in (0,ε) such that f(x)≥0. Does this mean that there is some δ>0 such that f(x)≥0 for all x in (0,δ)?

If true, how to prove it?

If it is not true, what would be a counterexample?

What additional assumptions on f would make it true?

I have been trying to prove this, but can't seem to discard the possibility of an oscillating f, with the oscillations taking place on smaller and smaller intervals and with lesser and lesser height as x=0 is approached. Still, intuition says that when x=0 is left, for example towards the right, f(x) must go somewhere.
 
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  • #2


No, it's not true. A counterexample would be
$$f(x) = \begin{cases}
x^3 \sin(1/x) & \textrm{ if }x \neq 0 \\
0 & \textrm{ if } x = 0\\
\end{cases}$$
It's easy to check that this function is continuously differentiable everywhere, and satisfies ##f(0) = f'(0) = 0##, and it takes on both positive and negative values on the interval ##(0,\epsilon)## no matter how small we make ##\epsilon##.

More generally, for ##n \in \mathbb{N}##,
$$f(x) = \begin{cases}
x^n \sin(1/x) & \textrm{ if }x \neq 0 \\
0 & \textrm{ if } x = 0\\
\end{cases}$$
will ##n/2## times differentiable if ##n## is even, but not ##(n/2)## times continuously differentiable. And it will be ##(n-1)/2## times continuously differentiable if ##n## is odd. When they exist, the derivatives at ##x=0## will be zero. And of course all of these functions will be counterexamples as well, due to the rapid oscillation as ##x \rightarrow 0##.

So it seems that, at minimum, you will need infinite differentiability. I'm not sure if there is a counterexample in that case.
 
  • #3


Wow, thanks, that's really useful. Would you agree that if additionally f'(x) changes sign a finite number of times, then the statement would hold?
 
  • #4


Skalman said:
Wow, thanks, that's really useful. Would you agree that if additionally f'(x) changes sign a finite number of times, then the statement would hold?
Yes, it should be no problem then. That would imply we can find an interval ##(0,\epsilon)## over which either ##f'(x) \geq 0## or ##f'(x) \leq 0##. Suppose the first inequality holds throughout ##(0,\epsilon)##. Then if ##0 < x < \epsilon##, we may apply the mean value theorem to conclude that
$$\frac{f(x)}{x} = \frac{f(x) - f(0)}{x - 0} = f'(y)$$
for some ##0 < y < x##. Therefore
$$f(x) = f'(y) x \geq 0$$
As ##x## was arbitrary in ##(0,\epsilon)##, we conclude that ##f \geq 0## on the entire interval.
 
  • #5


By the way, notice that with the new constraint, we didn't need ##f'## to be continuous, nor did we need ##f'(0) = 0##, nor even for ##f## to be differentiable at ##0##. We just need ##f(0) = 0## and the hypotheses for the mean value theorem: ##f## is continuous on ##[0,\epsilon]## and differentiable on ##(0,\epsilon)##.
 
  • #6


Ok, I see it very clearly now, thanks again for the help!
 

1. What is a differentiable function?

A differentiable function is a function that has a derivative at every point in its domain. This means that as you zoom in on any point on the graph of the function, it appears increasingly linear or flat.

2. How is a differentiable function related to continuity?

A differentiable function is always continuous, but a continuous function is not necessarily differentiable. This means that all differentiable functions are also continuous, but not all continuous functions are differentiable.

3. What is the significance of the neighborhood of x in a differentiable function?

The neighborhood of x in a differentiable function refers to the set of all points that are within a certain distance from x. This is important because the behavior of a differentiable function at a specific point is influenced by the values of the function in its neighborhood.

4. How does the derivative of a differentiable function change in the neighborhood of x?

The derivative of a differentiable function changes in the neighborhood of x because as you zoom in on a specific point, the slope of the function changes. This is because the function appears increasingly linear or flat, causing the derivative to approach a specific value at that point.

5. Can a differentiable function have a derivative of zero in its neighborhood of x?

Yes, a differentiable function can have a derivative of zero in its neighborhood of x. This means that at that specific point, the function is neither increasing nor decreasing, and is instead flat or constant. However, the function can still be changing in other parts of its neighborhood.

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