Question

Existence and Uniqueness Theorem for ODEs
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I know how to apply the theorem, but I was just wondering what the reasoning behind it was as an abstraction. I was thinking along the lines of "Here's a point in an interval, and in this interval, the derivative is continuous. I can sort of start out at an initial value point, and using the derivative, find out the next point on the curve, over and over again, until I finally have a solution to the ODE."

Does this make sense?