Exercise 6

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

If $\partial f/ \partial x$ is continuous on the rectangle $$R = \left\{ (t, x) : 0 \leq |t - t_0| \lt a, 0 \leq |x - x_0| \lt b \right\},$$ prove that there exists a $K \gt 0$ such that $$|f(t, x_1) - f(t, x_2) | \leq K |x_1 - x_2|$$ for all $(t, x_1)$ and $(t, x_2)$ in $R$.