Hello,
If $f:\mathbb{R} \to \mathbb{R}$ a differentiable function, it is very easy to find its Lipschitz constant. Is there any way to extend this to functions $f: \mathbb{R} \to \mathbb{R}^n$ (or similar)?
Hello, If $f:\mathbb{R} \to \mathbb{R}$ a differentiable function, it is very easy to find its Lipschitz constant. Is there any way to extend this to functions $f: \mathbb{R} \to \mathbb{R}^n$ (or similar)? 


Let $f = (f_1,\ldots,f_n): [a,b] \rightarrow \mathbb{R}^n$ be a continuously differentiable function. (See the comments above for an explanation as to why the hypotheses have been strengthened.) For $1 \leq i \leq n$, let $L_i = \max_{x \in [a,b]} f_i'(x)$, so that, by the Mean Value Theorem, for $x,y \in [a,b]$, $f_i(x)f_i(y) = f_i'(c)xy \leq L_i xy$. Then, taking the standard Euclidean norm on $\mathbb{R}^n$, $f(x)f(y)^2 = \sum_{i=1}^n f_i(x)f_i(y)^2 \leq (\sum_{i=1}^n L_i^2) \ xy^2$, so $f(x)f(y) \leq \sqrt{(\sum_{i=1}^n L_i^2)} \ xy$. Thus we can take $L = \sqrt{\sum_{i=1}^n L_i^2}$. Since all norms on $\mathbb{R}^n$ are equivalent  i.e., differ at most by a multiplicative constant  the choice of norm on $\mathbb{R}^n$ will change the expression of the Lipschitz constant $L$ in terms of the Lipschitz constants $L_i$ of the components, but not whether $f$ is Lipschitz. 


In fact a statement similar to what was described by Pete Clark is true for all normed vector spaces: Let $X$ and $Y$ be normed vector spaces. A (total) differentiable function $f:X\to Y$ is Lipschitz iff its derivative is bounded. Every upper bound for the differential is a Lipschitz constant. One direction follows from the mean value theorem: $\f(x)f(y)\\leq \Df(\xi)\\cdot\xy\$ for some $\xi$ on the straight line from $x$ to $y$. The other follows immediately from $Df(x)=\lim_{h\to 0}\frac{f(x)f(x+h)}{\h\}$ 


Another nice way to do this computation (which generally gives more precise information) is to use the formula $ f(x + h)  f(x) = [ \int_0^1 Df(x + th) dt ] h$ Also, the formula for $Df(x)$ in Hahn's post above is not correct. 

