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Note that Theorem 7.24 states only the existence of a solution of the differential Eq. (7.11) if $f$ is continuous. For uniqueness of the solution, we need a stronger condition. This subject will be handled later (see Theorem 9.30).

## 7.7* Banach Spaces and the Space ${C}^{1}\left(a\text{,}b\right)$

In this section we are looking to a suitable structure for a subset of $C\left(a\text{,}b\right)$, consisting of all differentiable functions with continuous derivatives ...

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