The constant c is an essential ingredient in this theorem. To see why it's important, look at this example: Let , defined for and . The initial value problem has the solution . But that is only defined on the interval [0,1), not on the interval [0,2] on which we defined f(x,t). Admittedly, the function f(t,x) does not explicitly involve t in that example, but it shows that the solution to the problem can blow up unexpectedly in the interval [a,b]. That is why it's going to be necessary to restrict to a shorter interval if you want the function x(t) to exist.

You can find a solution to the problem here (it's a theorem due to Peano). The proof is quite subtle. It starts by using the continuity of the function f(t,x) to get a neighbourhood of on which f is bounded.