It is well known that for iid random variables $X_1, \ldots, X_n$ with variance $\sigma^2$ that

$$\frac{1}{n-1} \sum_{i=1}^n (X_i - \overline X)^2$$

gives an unbiased estimator for $\sigma^2$, but

$$\sqrt{ \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline X)^2 }$$

is not an unbiased estimator for $\sigma$.

Is there a way to formulate precisely and prove that there does not exist an unbiased estimator $f(X_1,\ldots,X_n)$ for $\sigma$ that works for all probability distributions?