Determining smoothness of a function

[Unity+]

Given a function f(x) that is not a regular polynomial equation(x^n +/- x^n-1 +/- x-2...), how would one determine if a function is smooth over a curve or not?

 

For example, let us say there is a given function that has a curve involved. Given the conditions above, how would one determine if all the parts of the curve are smooth in the sense that there are no other irregular curves on that curve even at the most minuscule spot of the curve?

[John]

When you ask about "irregular curves" on the main curve, are you just talking about singular points? In any case, with the caveat that analysis isn't exactly my strong point, I don't know if there's a slick general way to determine whether any given function is smooth, and I wouldn't expect there to be such a method.

In some cases, smoothness or lack thereof is apparent, e.g. in the case of polynomials, sine and cosine, absolute value, the greatest integer function, etc. Simply taking derivatives may yield an obvious pattern and make smoothness (or not) obvious. Along these lines, I would imagine mathematical induction is useful at least some of the time as well, i.e. showing the function in question is continuous, and then showing that if the n^th derivative is continuous, then the (n + 1)^th derivative must also be continuous.

Edited May 7, 2014 by John

[Unity+]

When you ask about "irregular curves" on the main curve, are you just talking about singular points? In any case, with the caveat that analysis isn't exactly my strong point, I don't know if there's a slick general way to determine whether any given function is smooth, and I wouldn't expect there to be such a method.

 

In some cases, smoothness or lack thereof is apparent, e.g. in the case of polynomials, sine and cosine, absolute value, the greatest integer function, etc. Simply taking derivatives may yield an obvious pattern and make smoothness (or not) obvious. Along these lines, I would imagine mathematical induction is useful at least some of the time as well, i.e. showing the function in question is continuous, and then showing that if the n^th derivative is continuous, then the (n + 1)^th derivative must also be continuous.

I'm talking about where between [a, b], is the curve not "smooth"(where it follows a parabolic curve in a sense) or does it contain more wave-like structures?

 

Here is an example:

 

 

The problem with this and using the method of taking derivatives is if there is any point on the curve so minuscule, yet there is an irregularity then it may take a long time to detect this if it even exists on the curve.

 

There was this: http://en.wikipedia.org/wiki/Smooth_function, but it didn't seem to be the same kind of smoothness.

Edited May 7, 2014 by Unity+

[John]

Yes, by smooth, I'm taking you to mean "has derivatives of all orders." Do you mean "continuous" or something instead?

Edited May 7, 2014 by John

[Unity+]

Yes, by smooth, I'm taking you to mean "has derivatives of all orders." Do you mean "continuous" or something instead?

Derivatives of all orders seems to be the one I'm talking about.