# A continuously differentiable map is locally Lipschitz

Let $$f:\mathbb R^d \to \mathbb R^m$$ be a map of class $$C^1$$. That is, $$f$$ is continuous and its derivative exists and is also continuous. Why is $$f$$ locally Lipschitz?

### Remark

Such $$f$$ will not be globally Lipschitz in general, as the one-dimensional example $$f(x)=x^2$$ shows: for this example, $$|f(x+1)-f(x)| = |2x+1|$$ is unbounded.

• Well, $x \mapsto x^2$ is not particularly Lipschitz, is it? You seem to miss either a locally or that $df$ must be bounded.
– t.b.
Oct 17 '11 at 15:37
• @t.b. In some contexts $C^1$ implies that $\sup |f| + \sup |df| < \infty$. (At least, when I write $C^1(M,\mathbb{R}^k)$ with $M$ being non-compact, that's what I would mean.) Of course, this is not what the OP wrote in the parenthetical. Oct 17 '11 at 15:41
• yes you're right. I meant locally Lipschitz.
– bass
Oct 17 '11 at 15:41
• @bass: use that continuous functions are locally bounded. Oct 17 '11 at 15:43
• First reduce to the scalar-valued case (this is easy). Then note that $\left|f\left(x\right)-f\left(y\right)\right|=\left|\int_{0}^{1}\frac{d}{dt}f\left(tx+\left(1-t\right)y\right)dt\right|$. Then use the chain rule and...
– Mark
Oct 17 '11 at 17:20

If $f:\Omega\to{\mathbb R}^m$ is continuously differentiable on the open set $\Omega\subset{\mathbb R}^d$, then for each point $p\in\Omega$ there is a convex neighborhood $U$ of $p$ such that all partial derivatives $f_{i.k}:={\partial f_i\over \partial x_k}$ are bounded by some constant $M>0$ in $U$. Using Schwarz' inequality one then easily proves that $$\|df(x)\|\ \leq\sqrt{dm}\>M=:L$$ for all $x\in U$. Now let $a$, $b$ be two arbitrary points in $U$ and consider the auxiliary function $$\phi(t):=f\bigl(a+t(b-a)\bigr)\qquad(0\leq t\leq1)$$ which computes the values of $f$ along the segment connecting $a$ and $b$. By means of the chain rule we obtain $$f(b)-f(a)=\phi(1)-\phi(0)=\int_0^1\phi'(t)\>dt=\int_0^1df\bigl(a+t(b-a)\bigr).(b-a)\>dt\ .$$ Since all points $a+t(b-a)$ lie in $U$ one has $$\bigl|df\bigl(a+t(b-a)\bigr).(b-a)\bigr|\leq L\>|b-a|\qquad(0\leq t\leq1)\>;$$ therefore we get $$|f(b)-f(a)|\leq L\>|b-a|\ .$$ This proves that $f$ is Lipschitz-continuous in $U$ with Lipschitz constant $L$.
• I have a little doubt, isn't $df$ with respect to $t$ , then how can we justify the inequality ? Nov 18 '13 at 10:05
• @Theorem: $df\bigl(a+t(b-a)\bigr)$ is the derivative ("Jacobian") of $f$, evaluated at the point $a+t(b-a)\in U$. Nov 18 '13 at 10:14
• @kam:Consider the function $f(x):=x^2\sin(1/x^3)$. With $f(0):=0$ it is differentiable on all of ${\mathbb R}$, but it is not Lipschitz continuous near $x=0$. Mar 8 '20 at 9:55
Maybe this can help. The Lipschitz condition comes many times from the Mean Value Theorem. Search the link for the multivariable case. The fact that $f$ is $C^1$ helps you to see that when restricted to a compact set the differential is bounded. That's why you only have local Lipschitz condition.
A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition.The function $f(x) = x^2$ with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous.