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

[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.

 

[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 valuesIn 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

 

[AlfredPyo]

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

 

[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

 

[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?

 

[HallsofIvy]

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