 Lipschitz gradient

A differentiable function has a $\beta$-Lipschitz gradient on a set $D$ if its gradient satisfies Lipschitz continuity there:All norms here are the Euclidean norm. The constant limits how fast the gradient changes; it is not a bound on the magnitude of the gradient. For $f(x)=(x_1^2+9x_2^2)/2$, the gradient is the linear map $x\mapsto \mathrm{diag}(1,9)x$. Its operator norm is $9$, so $\beta =9$ works globally and is sharp by taking $x-y=(0,1{)}^T$. The gradient itself is unbounded on the whole Euclidean space.

If $f$ has a continuous Hessian matrix on an open convex set $D$ and $\Vert {\nabla }^{2}f(z){\Vert }_2\le \beta$ throughout $D$, integrate that linear map along the line segment from $x$ to $y$:The operator norm bound proves the displayed Lipschitz continuity. Convexity of the domain keeps the entire line segment inside it. Bounding the Hessian matrix only at the endpoints is insufficient.