Why does the continuity of Lyapunov function matter?
Is there any example where discontinuity of $V$ is problematic even if it does not change the sign but it is strictly decreasing and yet not stable?
From wikipedia:
A Lyapunov function for an autonomous dynamical system
\begin{cases}g:\mathbb {R} ^{n}\to \mathbb {R} ^{n}\\{\dot {y}}=g(y)\end{cases}
an equilibrium point at $y=0$ is a scalar function
$$V: \mathbb{R}^n\to \mathbb{R}$$
that is continuous, has continuous derivatives, is locally positive-definite, and for which $-\nabla V.g$ is also locally positive definite. The condition that $-\nabla V.g$ is locally positive definite is sometimes stated as $\nabla V.g$ is locally negative definite.