# What does it mean for a function to be defined on an interval?

1. May 11, 2014

### AlfredPyo

So I was looking at one of the definitions of first order DE's.
But I don't get what this statement means:

let a function f(x) be defined on an interval I.

2. May 11, 2014

### lurflurf

I assume you are working with real numbers.
A (real) interval is a convex subset of the real numbers.
So I is some subset of R such that
$$\text{if }x_k,a_k\in \mathbb{R} \text{ and }x_k\in I \text{ with}\sum_{k=1}^n a_k= 1 \text{ then }\sum_{k=1}^n x_k a_k \in I$$
In other words we have a block of values for which the function is defined for all values

In differential equations we can have problems if the function is defined on a set with gaps such as in the set of integers or rationals.

Things like
$$x\le 7\\ x>57\\ 3<x\le 7 \\ x\in \mathbb{R}\\ x\in \emptyset\\ x=1$$
are what you should have in mind

Last edited: May 11, 2014
3. May 11, 2014

### AlfredPyo

Ok, so basically to be defined means to not have any discontinuities, right?

4. May 11, 2014

### lurflurf

^There can be discontinuities, there cannot be gaps.
$$\mathrm{f}(x)=\begin{cases}\phantom{\frac{0}{0}}0 & x<0 \\ \,\,\,\, \frac{1}{2} & x=0\\ \phantom{\frac{0}{0}}1&x>0\end{cases}$$
is defined on an interval, but not continuous

5. May 11, 2014

### AlfredPyo

Ok, so a piecewise function. As long as in the interval, all x values in the interval has one y value or vice versa?

6. May 12, 2014

### HallsofIvy

Staff Emeritus
That's pretty much the definition of "function", isn't it?