What is the largest possible ? in Example 1 in the text?

Data from example 1

Consider the initial value problem

and take the rectangle R; |x|

This solution is discontinuous at + ?/2, and there is no continuous solution valid in the entire interval |x|

The conditions in the two theorems are sufficient conditions rather than necessary ones, and can be lessened. In particular, by the mean value theorem of differential calculus we have

where (x, y₁) and (x, y₂)are assumed to be in R, and y^? is a suitable value between y₁ and y₂. From this and (3b) it follows that

It can be shown that (3b) may be replaced by the weaker condition (4), which is known as a Lipschitz condition. However, continuity of f(x, y) is not enough to guarantee the uniqueness of the solution. This may be illustrated by the following example.

y' = 1 + y, y(0) = 0