For the standard definition of Lipschitz continuity we have:
$$||f(x)−f(y)||≤L||x−y||$$ s.t. $x,y \in R^n$
In case the function $f(x)$ maps to $R$, the Lipschitz constant can be estimated by $max \nabla(f(\psi))$, where $\psi$ is a point in the line connecting the two points $x$ and $y$. What if $f(x)$ maps to $R^m$? Is there an equivalent method to estimate $L$?
If $f$ is differentiable and its Jacobian matrix $J$ satisfies $\| J \| \le M$ on a curve $\gamma(t)$, $a \le t \le b$, of length $L$ (where $\|\cdot\|$ is the operator norm corresponding to the Euclidean norms on $\mathbb R^n$ and $\mathbb R^m$), then $$ \eqalign{\|f(\gamma(b)) - f(\gamma(a))\| &= \left\|\int_a^b \dfrac{d}{dt} f(\gamma(t))\; dt \right\|\cr &\le \int_a^b \left\| \frac{d}{dt} (f(\gamma(t))) \right\|\; dt\cr &= M \int_a^b \|\gamma'(t)\|\; dt = M L}$$ In particular, if the curve is a straight line, we find that the Lipschitz constant is bounded by the maximum of $\|J\|$.