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I am reading about sensitivity equations. In the book by Khalil they write

Khalil

and go on to derive the sensitivity equations.

My question is: how do you go about proving joint coninuity in $(t,x,\lambda)$ for a given ODE-system? For example, in this example $$\dot{x_1} = x_2 = f_1(t,x,c) $$ $$\dot{x_2} = -c sinx_1 = f_2(t,x,c)$$

I would have to show $lim_{(t,x,c) \to (t_0,x_0,c_0)} f_1(t,x,c) = f_1(t_0,x_0,c_0)$ (for all $(t_0,x_0,c_0)$) and the same for $f_2$. I do not understand how to show this or how to even start and I cannot find any literature on it. What am I missing?

Khalil, Hassan K., Nonlinear systems., Upper Saddle River, NJ: Prentice Hall. xv, 750 p. (2002). ZBL1003.34002.

ari
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    I thought seperate continutiy does not imply joint continuity (see here https://math.stackexchange.com/questions/98831/continuity-and-joint-continuity). – ari Sep 16 '22 at 12:29
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    Composition of continuous (differentiable) functions and operations gives again a continuous (differentiable, etc.) function. This covers most examples. The limit or open-set definition only needs to be applied for theoretical purposes or borderline singular examples. – Lutz Lehmann Sep 16 '22 at 12:35

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