Using Bolzano's theorem and studying the derivative I could prove that, for all $a\in\mathbb{R}$, the equation $e^x+x^3+x+\cos x = a$ has an unique solution. Let's call it $x_a$ and define the function $g:\mathbb{R}\rightarrow\mathbb{R}$ as $g(a)=x_a$, that is, the function that to each $a\in\mathbb{R}$ associates the unique solution of $e^x+x^3+x+\cos x = a$. I'm wondering if $g$ is continuous. I tried to prove by the characterization $\varepsilon-\delta$ of the continuity, but I didn't end up with something useful. Any help or advice?
Thanks in advance.