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Let $C := \{ g: [a,b] \times \mathbb{R} \rightarrow \mathbb{R} | a, b \in \mathbb{R}, a < b \text{ and } g \text{ is derivable} \} $

Let $f: [0,\infty[ \times \mathbb{R} \rightarrow \mathbb{R}: (z,u) \mapsto f(z,u)$ a derivable function in all variables.

Let $g: [\Delta t, \infty[ \rightarrow C: t \mapsto g_t$ with $g_t: [t-\Delta t, t+\Delta t] \times \mathbb{R} \rightarrow \mathbb{R}: (t',u) \mapsto f(t',u)$

How can we calculate $\dfrac{\partial g}{\partial t}$ ?

I don't have a clue how to calculate this. Can someone help me ?

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    What notion of limit in the function space are you using? It suffices to define a norm on $C$. – 2017-01-19
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    You should also explain why $C$ is a linear space. Since $a$ and $b$ depend on $g$, I am not sure how to add two functions with a different domain, i.e. what is the sum of the characteristic functions $\chi_{[0,1]}$ and $\chi_{[2,3]}$? – 2017-01-19
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    @gerw If I am correct, if I make every closed interval open, this isn't a problem anymore, because to calculate the derivative you only need to know the difference in the limit point – 2017-01-19
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    And what is the sum of the above characteristic functions? Even with open intervals, the sum will never be differentiable. – 2017-01-19

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