Is there any rule that says that this range cannot be $\mathbb{R}$? Apparently there is, but then what about functions like $\log |x|$?
2026-02-23 20:38:12.1771879092
The range of a continuous even function
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Consider $f(x) = x\sin x$. It is easy to check that $f$ is even.
Also, $f$ is continuous and obtains arbitrarily large positive and negative values. You can use the intermediate value theorem to show that every real number is in the range of $f$.
Hence, $f$ is a continuous even function whose range is $\mathbb{R}$. Therefore, there is no rule that a continuous even function cannot have all of $\mathbb{R}$ as its range.