“Good” functions
What pops into your mind when someone mentions a “good”, or equally vague, but more acceptable “well-behaved” function? What are the conditions a function must satisfy to be well-behaved? Depending on the context, it may be any of an uncountable number of things: bounded variation, differentiability, smooth, bounded, continuous, lipschitz continuous, one-to-one, etc. But if you’re like me, if someone shouted out “that’s a good function, buddy!”, you would think of a collage of all of these things.
And if you’re like me, you converged on your personal truth of what makes a function a “good” function through a process of iteration. First, and for a long time, it was continuity; but then, you realized a lot of functions (piecewise defined ones, for example), are continuous, but not nice. Then, and for a longer time, it was differentiability. Why doesn’t this serve as a satisfactory definition of a good function? Well, it could, if you were willing to admit functions whose have non-convergent Taylor series. But, you probably wouldn’t, and there are lots (of course?, since there is one) of functions which are differentiable, but have non-convergent taylor series.
Also, in my intuitive conception of a good function, it is impossible for a good function to be identically zero on an interval, unless that good function is itself the zero function. I thought, until today, that it could be proven that that condition is a necessary result of a function being differentiable. Too bad; that’s not true, at all.
An example of a differentiable function with a non-convergent Taylor series: the Cauchy function, defined piecewise as 
, is in fact continuous and differentiable at all 
; since 
, the T.S. expansion taken at 
is identically 0.
“Mollifiers” are examples of differentiable functions that are identically zero on an interval, yet are not identically the zero function. One mollifer is the piecewise function 
, which is a bump in the first interval of definition, and zero everywhere else.
This all raises a question: is it possible to have a continous differentiable function which has a non-convergent T.S., and/or is zero on an interval yet is not identically zero, which is not piecewise defined?
The search continues for a satisfactory standard for a “good” function. I hear tell of analyticity, but have no idea of what that means yet…
Possibly relevant posts:
- Complex Analysis pt. I (11/3/2003)
- Approximate Identities (7/2/2005)
- Convexity (2/28/2005)