is any filter whose impulse response has a discontinuity in the first derivative fundamentally high-pass?
This actually isn't the case. Consider the continuous-time triangle function:

$$
\text{tri}(t) = \begin{cases}
1 - |t|, &amp;&amp; |t| &lt; 1 \\
0, &amp;&amp;\text{otherwise}
\end{cases}
$$

This function has the Fourier transform:

$$
\text{Tri}(f) = \text{sinc}^2(f)
$$

This follows readily from the fact that the triangle function can be constructed by convolving a rectangle function, which has a Fourier transform of $\text{sinc}(f)$, with itself. The $\text{sinc}$ function decays to zero as its argument approaches infinity (and its square decays even faster yet), so the triangle window does not have a highpass characteristic.