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Given a set $B=\{x | x:[t_0, T]\rightarrow \mathbb{R}^n,$ $x$ is a differentiable function on $[t_0, T]-\{t_1,...,t_M\}$ and $x(t^{+}_k)=\lim \limits_{t \rightarrow t^{+}_k} x(t)$ and $x(t^{-}_k)=\lim \limits_{t \rightarrow t^{-}_k} x(t)$ exists with the condition $x(t^{-}_k)=x(t_k)$ and $x(t_0)=\lim \limits_{t\rightarrow t^{-}_0}x(t)\}.$ Then my question is whether $B$ is a banach space on real field? Further whether it means all differentiable functions on $[t_0, T]$ will be included in $B?$

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    What norm are you using? – 2017-02-14
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    Clearly, $B$ is closed under linear combinations (though I think you mean that $x(t_{0})=\lim_{t\rightarrow t_{0}^{+}}x(t)$), so the next important question is what the norm would be on such a space. Then, we would need to verify whether or not $B$ is complete with respect to this norm. – 2017-02-14
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    $\|x\|=\sup \limits_{t\in [t_0, T]} \|x(t)\|$ where $\|x(t)\|$ is an Euclidean norm on real space $\mathbb{R}^n$ – 2017-02-14
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    No, it's not a Banach space, as you could approximate a non-differentiable function by differentiable ones. – 2017-02-14
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    If i take x to be almost everywhere differentiable function on $[t_0, T]-\{t_1,...,t_M\}?$ – 2017-02-14

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