Not quite a discontinuity?

In summary, the conversation discusses a function that is defined as the greater of x and x2, which creates a straight line graph before turning into a curve. This function is continuous but not smooth, meaning it is not differentiable. It is referred to as a "piecewise" function and can be described as "piecewise smooth". There are also other functions, such as the Weierstrass function, that are continuous but not differentiable anywhere. The term "smooth" requires derivatives of all orders to exist.
  • #1
Jehannum
102
26
For example, if we define f(x) as "the greater of x and x2" it will give a straight line graph between (0,0) and (1,1) then turn into a curve. This function is continuous but not 'smooth'.

Is there any special name for this kind of function?

Are there any interesting considerations about such functions - or is it just a case of 'split them up into parts when necessary'?
 
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  • #2
"Continuous but not differentiable"?
There are functions continuous everywhere, but not differentiable anywhere (e. g. the Weierstrass function).
Smooth is a mathematical term, and requires derivatives of all orders to exist.
 
  • #3
Jehannum said:
Is there any special name for this kind of function?

The adjective "piecewise" could be used. (https://en.wikipedia.org/wiki/Piecewise ). You could say the function is "piecewise smooth".
 
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Question 1: What is meant by "not quite a discontinuity" in science?

"Not quite a discontinuity" refers to a phenomenon in science where there is a sudden change or jump in a system or process, but it is not a complete or perfect break. This means that there is still some connection or continuity between the two sides of the change, rather than a complete separation.

Question 2: How is "not quite a discontinuity" different from a true discontinuity?

A true discontinuity is a complete break or separation between two states in a system, while "not quite a discontinuity" is a partial or incomplete break. This means that there is still some connection or continuity between the two sides of the change in "not quite a discontinuity".

Question 3: What are some examples of "not quite a discontinuity" in science?

One example of "not quite a discontinuity" is the phase transition between liquid and gas states, where there is a sudden change in properties but the two states are still connected. Another example is the transition between different crystal structures in solid materials.

Question 4: How do scientists study "not quite a discontinuity"?

Scientists study "not quite a discontinuity" using various experimental and theoretical methods. These can include analyzing data and observations, conducting experiments, and creating mathematical models to understand the underlying mechanisms behind the phenomenon.

Question 5: Why is understanding "not quite a discontinuity" important in science?

Understanding "not quite a discontinuity" is important in science because it helps us to better understand and predict changes in natural systems. It also allows us to develop new technologies and materials that take advantage of these types of transitions.

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