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This paper addresses the Mountain Pass Theorem for locally Lipschitzfunctions on finite-dimensional vector spaces in terms of tangencies. Namely,let $f \colon \mathbb R^n \to \mathbb R$ be a locally Lipschitz function with amountain pass geometry. Let $$c := \inf_{\gamma \in \mathcalA}\max_{t\in[0,1]}f(\gamma(t)),$$ where $\mathcal{A}$ is the set of allcontinuous paths joining $x^*$ to $y^*.$ We show that either $c$ is a criticalvalue of $f$ or $c$ is a tangency value at infinity of $f.$ This reduces to theMountain Pass Theorem of Ambrosetti and Rabinowitz in the case where thefunction $f$ is definable (such as, semi-algebraic) in an o-minimal structure.

The mountain pass theorem in terms of tangencies