If $f$ and $\dfrac{\partial f}{\partial y}$ are continuous in a rectangle $R: |t| \leq a, |y| \leq b,$ then there is some interval $|t| \leq h \leq a$ in which there exists a unique solution $y = \phi (t)$ of the initial value problem, $y^{'} = f(t,y)$
The book doesn't provide a proof of this theorem and I wanted to know why this is true. I know continuity implies that $\int f(t,y)$ exists but from there how do you show there is a smaller interval and or rectangle and that the solution is unique.
See the Picard-Lindelöf theorem (the general version has slightly weaker hypotheses than yours)