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How to determine the value of an $a \in \mathbb{R}$ parameter so that there exists an antiderivative for the $f:\mathbb{R} \Rightarrow \mathbb{R}$, $$f(x)=\begin{cases}\sin\frac{e^x}{x},& x\neq 0\\ a, &x=0\end{cases}$$ piecewise function.

My first idea would be to transform $\sin\frac{e^x}{x}$ into the sum or difference or maybe even the product of two elementary functions and from there I only need to check that the two functions are continous in the $x=0$ point. Now this sounds great in theory but can I transform $\sin\frac{e^x}{x}$ in such a way?

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  • $\begingroup$ I'm guessing that the term "antiderivative" is used in this sense: We want a function $F$ so that $F'(x)=f(x)$ everywhere, including $x=0$). From this it does not follow that $f$ is continuous. For example, $f_1(x) = \sin(1/x)$ does have such an antiderivative. $\endgroup$
    – GEdgar
    Commented Dec 11, 2023 at 18:49

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The antiderivative of a function doesn't depend on its value at any specific point - the value you assign at $a$ won't make a difference.

You can also see that there's no value you can assign at 0 that will make this function continuous. As $x$ approaches 0, $e^x$ approaches $1$ and $\frac 1x$ approaches $\pm \infty$. In particular, $\frac {e^x}{x}$ will approach $\pm \infty$ as $x$ approaches $0$ from the right and left respectively. Then $\sin(\frac {e^x}{x})$ will oscillate between $\pm 1$, so no limit can exist at 0.

This isn't a priori a problem: an antiderivative does exist, but such an antiderivative probably can't be expressed in terms of elementary functions.

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