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I have been reading about existence and uniqueness of solutions to linear ODE's but I believe this question isn't exclusively related to them.

There was a theorem I was looking at which was about proving the existence of a unique function on some interval $I=(x-R,x+R)$ for some $R>0$ and more specifically it talked about continuity on this interval.

My question is this, I really don't understand what it means when a solution exists on an interval, an interval is just a set of numbers around a point so we are talking about a collection of numbers but when I think of a solution to an ODE I'm thinking about a function that may not be a constant number. So something like $y(x)=x^2$ for example, how exactly does this relate to the interval? Or is the interval saying the function that satisfies the differential equation is only defined for numbers that lie in that interval. And what would it mean to be a unique solution on an interval? The only point in an interval where a function that satisfies the ODE is defined? Or is the uniqueness referring to the function itself.

I guess I have got myself in a bit of a mess here by not understanding what is meant by these terms and I don't see how I can progress any further in understanding DE's without getting to grips with what is meant by these words in this context.

Clarification would be brilliant thank you.

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You’ve got it: The function that satisfies the equation is only defined for numbers in the given interval, and uniqueness refers to the function, i.e., of all functions that might be defined for numbers in the interval, only that function satisfies the equation.

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