Operator norm and Lipschitz continuous problem

Question

Suppose that $f : \mathbb R^6 \rightarrow \mathbb R$ is a function with the following two properties: $f(0) = 0$, and at at any point $c \in \mathbb R^6$ and any increment $h$, $\Vert Df(c)(h)\Vert \le \Vert h\Vert $. Show $f(B_1(0)) \subseteq (-1, 1)$. It says use mean value theroem at some point. An interpretation is if the operator norm of $Df$ is at most 1 at all points then $f$ is Lipschitz continuous. I cannot find operator norm or Lipschitz continuous in my notes/textbook so please explain what those are too!

Answer 1

The mean value theorem implies that for each $h$ there exists $\lambda \in [0, 1]$ (depending on $h$) such that $$f(h) = f(0) + Df(\lambda h)(h).$$ In this case, $f(0) = 0$, so we have $$|f(h)| = |Df(\lambda h)(h)| \le \|h\|.$$ Thus if $h \in B_1(0)$, then $f(h) \subseteq (-1, 1)$.

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Regarding the "interpretation": the operator norm of a linear operator $G$ is $\sup_{h \ne 0} \frac{\|G(h)\|}{\|h\|}$. Thus the statement "$\|D f(c) (h)\| \le \|h\|$ for any $h$" is equivalent to "the operator norm of $Df(c)$ is at most $1$."