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Construct an example of a differential equation depending on a parameter $a$ for which some solutions do not depend continuously on $a$.

i was reading differential equations, dynamical systems and introduction to chaos - Hirsch book and, in the chapter 7, i did not find any counterexample, please can anyone help me whit this? i was thinking this for 3 days, Thanks!

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Consider $y'+a^xy=0$. What happens if $a=1$? And if $a \in (0,\infty)\setminus\{1\}$?

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  • $\begingroup$ ok, i get it, the solution when $a=1$ is $y(t)=ce^{-x}$, and the solution when $a \neq 1$ is $y(t)=ce^{a^t/lna}$ and that solution is not continous if $a \leq 0$ is correct? but, then, we need to take $a \in (-\infty, \infty)$ or not? $\endgroup$ – P3peM4th. Mar 31 '17 at 13:56
  • $\begingroup$ When $a=1$ we have that $\ln a=0$, and $\ln a$ shows up in the denominator of the exponent of the solution to the case $a\neq 1$. $\endgroup$ – Fimpellizieri Mar 31 '17 at 17:58

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