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$$
\begin{aligned}
\lim _{x \rightarrow 0} f^{\prime}(x) &=\lim _{x \rightarrow 0} \frac{\left[g^{\prime}(x)+\mathrm{e}^{-x}\right] x-\left[g(x)-\mathrm{e}^{-x}\right]}{x^{2}} \\
&=\lim _{x \rightarrow 0} \frac{g^{\prime}(x)+\mathrm{e}^{-x}}{x}-\lim _{x \rightarrow 0} \frac{g(x)-\mathrm{e}^{-x}}{x^{2}} \\
& \stackrel{L^{\prime}}{ } \lim _{x \rightarrow 0}\left[g^{\prime \prime}(x)-\mathrm{e}^{-x}\right]-\lim _{x \rightarrow 0} \frac{g^{\prime}(x)+\mathrm{e}^{-x}}{2 x} \\
&=g^{\prime \prime}(0)-1-\frac{1}{2}\left[g^{\prime \prime}(0)-1\right] \\
&=\frac{1}{2}\left[g^{\prime \prime}(0)-1\right]=f^{\prime}(0),
\end{aligned}
$$
故 $f^{\prime}(x)$ 在 $(-\infty,+\infty)$ 上连续.
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