The main results of Section \[sec:singularity\] are summarised in Theorem \[thm:regularity II\] about the regularity of the highest wave. Lastly, we revisit in Section \[sec:global\] the bifurcation analysis from [@EK08; @EK11], ultimately proving that there is a sequence of waves converging to a wave of greatest height $\varphi(0) = \frac{\mu}{2}$, with a nontrivial wave speed $\mu \in (0,1)$. We underscore that several parts of Section \[sec:global\] are new with respect to [@EK08; @EK11], including the bifurcation formulas given in the proof of Theorem \[thm:local\_whit\].

Finally, the existence of a highest, cusped travelling-wave solution of the Whitham equation was announced earlier this year in [@E15_highest] (without proofs). The paper at hand provides a complete account of this fact, as well as several improvements—the most eminent examples being the regularity of the highest wave, and the properties of the kernels $K$ and $K_P$.

Completely monotone functions and the integral kernel $K$ {#sec:K}
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In this section we investigate the properties of the integral kernel $K$ in . Two routes towards understanding this transform are described—via positive definite functions (related to the Fourier transform), and via completely monotone functions (related to the Laplace transform). We start our exposition with a survey and analysis 