## Cauchy Problem with two solutions

Suppose is continuous and . Prove that if the Cauchy Problem has two distinct solutions then it has infinitely many solutions.

**Proof:** Without loss of generality we may assume that . Then there exist two solution of the Cauchy problem such that they are different in a point which we may assume is greater than . Therfore assume and denote a point on the segment we can extend towards . From Cauchy’s existence theorem, we can see that there exists a solution around of the differential equation . Denote by the compact determined by and the line . Since is in that compact in a left neighborhood of , by the compact extension theorem, it can be extended until it reaches the boundary of . From the intersection point of the graph of with the boundary of we can go on the graph of or until we reach and by the corollary of Lagrange’s theorem the graph we choose is the graph of a solution to the initial differential equation.

Therefore, we have found a solution for all , and therefore the intial equation has uncountably infinitely many solutions.