Idea about single-point differentiability and continuity

In summary, the most conventional definition of higher derivatives requires that the function exists in a neighborhood of the point of interest and the limit process for that function also exists in that neighborhood. This means that the derivative of previous order must exist on some interval instead of just a single point.
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hilbert2
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About function that are continuous or differentiable at isolated points.
Many have probably seen an example of a function that is continuous at only one point, for example

##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{R}\setminus\mathbb{Q}\end{array}\right.##

It is also possible to define a function for which the derivative defined as

##f'(x) = \underset{h\rightarrow 0}{\lim}\frac{f(x+h)-f(x)}{h}##

exists at only the point ##x=0##. One of the simplest examples is

##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x^2 , & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x^2 , & \hspace{6pt}when\hspace{3pt}x\in\mathbb{R}\setminus\mathbb{Q}\end{array}\right.##

for which the derivative at the origin is zero. Extending this to higher derivatives is not as easy, because the second derivative is usually seen as the same finite difference applied to the first derivative:

##f''(x) = \underset{h\rightarrow 0}{\lim}\frac{f'(x+h)-f'(x)}{h}##

and it doesn't make much sense to start calculating this for a function that has only ##x=0## as its domain.

However, the second derivative of

##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}|x|^3 , & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -|x|^3 , & \hspace{6pt}when\hspace{3pt}x\in\mathbb{R}\setminus\mathbb{Q}\end{array}\right.##

at ##x=0## could be calculated by the central finite difference

##f''(0) = \underset{h\rightarrow 0}{\lim}\frac{f(-h)-2f(0)+f(h)}{h}##

and found to have value zero.

A function with derivatives of arbitrary order being zero at ##x=0## and existing nowhere else could be defined by saying, e.g. ##f(0)=0## and ##f(x)=\pm\exp (-1/|x|)## when ##x\neq 0##.

Does the most conventional definition of 2nd and higher derivatives require that the derivative of previous order exist on some interval instead of just a single point?
 
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  • #2
hilbert2 said:
Does the most conventional definition of 2nd and higher derivatives require that the derivative of previous order exist on some interval instead of just a single point?
Yes. Differentiation is a local process. It requires all functions to exist in a neighborhood of the point. The limit process has to exist, i.e. all limits for ##t## in a neighborhood of ##t=0##.
 

1. What is single-point differentiability?

Single-point differentiability refers to the property of a function where the derivative exists at a single point. This means that the function is smooth and continuous at that specific point, and the slope of the tangent line can be defined at that point.

2. How is single-point differentiability different from overall differentiability?

Overall differentiability refers to the property of a function where the derivative exists at every point in its domain. Single-point differentiability only requires the derivative to exist at a single point, while overall differentiability requires it to exist at every point.

3. What is the relationship between single-point differentiability and continuity?

If a function is single-point differentiable at a specific point, it must also be continuous at that point. This is because the existence of a derivative at a point implies that the function is smooth and continuous at that point.

4. Can a function be continuous but not single-point differentiable?

Yes, it is possible for a function to be continuous at a point but not single-point differentiable. This can occur when the function has a sharp corner or a cusp at that point, where the slope of the tangent line is undefined.

5. How is single-point differentiability used in real-world applications?

Single-point differentiability is used in many areas of science and engineering, such as physics, economics, and computer science. It allows us to analyze the behavior of functions and make predictions about their rates of change at specific points, which is crucial in understanding and solving real-world problems.

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