How to make sense of this function in this proof of the Picard-Lindelöf theorem?

2018-07-22 02:48:14

See the pages below. I'm having trouble with the function $L_1$ defined at start of the second page. I have few questions:

On the 4th inequality in the second page, the author assumes that $L_1'(s) = L(s)$, but, without any other information, we can only do this if we know that $L$ is continuous (this is a condition in the Fundamental Theorem of Calculus). How do I show that $L$ is continuous?

If $L$ is continuous, isn't (2.26) automatically true?

This is sort of unrelated to the question, but $K: X \to X$ is defined by

$$

(Kx)(t) = x_0 + \int_{t_0}^t f(s, x(s))\ ds.

$$

Thanks in advance.