# Why are "smooth entropic quantities" useful/necessary?

Consider the $$\epsilon$$-smoothed relative max-entropy of $$\rho$$ with respect to $$Q$$, defined as (following Watrous' notation from these notes): $$\mathrm D_{\rm max}^{\epsilon}(\rho\|Q) = \min_{\xi\in B_\epsilon(\rho)} \mathrm D_{\rm max}(\xi\|Q),\tag1$$ where $$\rho$$ is a state, $$Q$$ a positive semidefinite operator in the same space, and $$B_\epsilon(\rho)$$ is the ball of states $$\epsilon$$-close to $$\rho$$ (e.g. in trace distance).

Similarly, (Tomamichel 2015) defines the "$$\epsilon$$-smooth max-entropy" as $$H_{\rm max}^\epsilon(X)_\rho \equiv \min_{\tilde\rho} H_{1/2}(X)_{\tilde\rho}, \tag2$$ where $$H_{1/2}(X)$$ is the Renyi entropy of order $$1/2$$, and the minimisation is performed over all states that are $$\epsilon$$-close to $$\rho$$ in trace distance. In page 6, they show that $$H_{\rm max}^\epsilon(X)$$ characterises the single-shot compressibility of a source $$X$$, in that given an error threshold $$\epsilon$$, the minimum possible $$m\equiv m^*(\epsilon)$$ such that there is an $$(\epsilon,m)$$-code for $$X$$ is bounded by $$H_{\rm max}^\epsilon(X)_\rho \le \log_2 m_*(\epsilon) \le \inf_{\delta\in(0,\epsilon)}\left[ H_{\rm max}^{\epsilon-\delta}(X)_\rho + \log_2(1/\delta) \right].$$

What I'm trying to understand better is why the operation of "smoothing" probability distributions/states works well to characterise single-shot compressibility/transmission rates/etc.

In the notes linked above, in page 26, Watrous writes that

The idea is that the smoothed max-relative entropy reflects a tolerance for small errors, which we often have or would like to express when analyzing operationally defined notions. Without smoothing, the max-relative entropy can sometimes, in certain settings at least, have unwanted hyper-sensitivities that smoothing eliminates.

This hints towards what I'm trying to get at, I think: min-max entropic quantities seem to not always behave nicely with respect to smooth changes of probability distribution/state, and thus we fix this by "smoothing" them. What are simple examples where this problem is evident, and then fixed via smoothing? Or more generally, is there a more precise or specific argument to see why smooth entropic quantities should be useful (as in, I can see that they are useful, but what specifically causes them to be so useful?)