# Does every curve have a function?

1. Feb 23, 2015

### thetexan

Let's say I want to calculate some instantaneous speeds during a trip from city a to city b.

I start down the highway from an initial speed of zero then increased to 55. After 15 minutes I increase to 70. Down to 60 for 2 minutes. Then up to 80 for 15. Back down to 70. Then a slow decrease to 0 for the last 5 minutes.

I keep all of the data points along the way and can take these data and plot a curve.

Is there a formula that represents this curve? How can I discover the formula? In other words how can a curve be reverse engineered to find the formula?

Which brings up the big question...does any simple curve I can draw on the blackboard have a formula and is there a way to find it?

Tex

2. Feb 24, 2015

### SteamKing

Staff Emeritus
Your description of how the speed changes is pretty vague, mathematically speaking. For example, "I start ... from an initial speed of zero then increased to 55 ..."

First of all, a speed of 55 what? Mph, kmh, furlongs / fortnight?
Second, how is the speed increasing? Over what time period is it increasing?

If you have speed-time data, there are several techniques you can use to construct a curve containing these points. There may not be a unique curve which fits, but the more data you have, the easier it will be to construct a curve.

3. Feb 24, 2015

### axmls

It is interesting to note, however, that given any finite set of n + 1 data points, there exists exactly one polynomial of degree n which goes through them all.

In the OP's case, it seems to be something that's best done with a little more information and a piecewise function.

4. Feb 24, 2015

### jbriggs444

Probably not what you are after but...

A curve that you draw on the blackboard, assuming an infinitely thin line drawn by ideal chalk, passes through an infinite set of points. The cardinality of this set of points (roughly speaking, the number of points on the curve) is the cardinality of the continuum. It is the same as the cardinality of the set of real numbers. This cardinality is denoted by $|\mathbb{R}|$. Naively, the number of curves that could be drawn is $|\mathbb{R}|^{|\mathbb{R}|}$ where the exponentiation is cardinal exponentiation.

However, any curve that you can draw on the blackboard without lifting the chalk must be continuous. A continuous function can be characterized completely by its value for all inputs that are rational numbers. The cardinality of the set of rational numbers turns out to be the same as the cardinality of the set of natural numbers, $|\mathbb{N}|$. So the number of continuous curves is given by $|\mathbb{R}|^{|\mathbb{N}|}$. Under the rules of cardinal exponentiation this turns out to be equal to $|\mathbb{R}|$ -- the cardinality of the set of real numbers.

The set of real numbers is "uncountable". That means that there is no way to make a list of the real numbers -- no way to put them in one to one correspondence with the set of natural numbers so that there is a first real number in the list, a second real number, a third and so on that covers all of the real numbers.

The set of formulas that you can define is "countable". There is a way to make a complete list of all possible formulas. For instance, sort them by length shortest to longest and within each formula length, sort them in alphabetical order.

If you could find a formula for each curve, that would mean that the set of curves would be countable. Since the set of curves is uncountable, it follows that you cannot find a formula for every curve. Some curves must have no corresponding formula.

5. Feb 24, 2015

### Staff: Mentor

6. Feb 24, 2015

### timthereaper

Second @axmls 's answer. I would add that you could make a spline from all the data points if all you're after is a curve. The hard part about "reverse-engineering" the formula is that the math has to make some sort of physical sense. If you knew something about the system you're taking data points from, you'll be able to get a better idea of what that curve actually means.

7. Feb 24, 2015

### thetexan

Well the point of my question is really this...can any curve (and let's just stick with a winding curve that doesn't cross itself) be represented by a formula?

Let's say I meticulously observe some phenomena and collect enough data to fairly represent the phenomena with a graphic curve. Can I also, in any and all cases, determine or create a formula that gives me that same curve, thus describing the phenomena, whatever it is, mathematically by that formula?

Tex

8. Feb 24, 2015

### jbriggs444

My argument for that point was based on an implicit assumption that the curve in question was arbitrary rather than one selected by a physical phenomenon and on an assumption that we were going for an exact match rather than an approximate match.

If one is after an exact match, and is restricting attention to curves resulting from physical phenomena rather than completely arbitrary continuous curves then the answer is "we do not know". We do not (and as far as we know, can not) know how tightly the physical universe restricts the set of available curves. It is not even clear that there is such a thing as an ideal curve in the real world.

If one is willing to accept an approximate match and if one accepts that all physical phenomena are continuous then I believe that the answer is "yes" -- one can come up with a function that matches any real curve within any desired non-zero error bound over any desired finite interval. Unless I am mistaken, this follows trivially because every continuous function on a compact domain is uniformly continuous. Worst case one can simply define the function piece-wise as a table of values and achieve any desired accuracy over any given closed interval.

9. Feb 24, 2015

### thetexan

My original example of a road trip was an example.

I make the trip and collect all data recording speeds, times between changes, etc. so that I can draw a graph representing the trip.

Now to be able to find speeds at any given instant during his trip I need to differentiate a formula that represents that data. But first I need a formula so I can find a derivative.

Tex

10. Feb 28, 2015

### HallsofIvy

I think that, before a meaningful answer can be given, we need to know what you mean by "function". What definition of "function" are you using?

11. Mar 14, 2015

### Alexandre

Nope you can't. Sorry for that, math is not omnipotent.