# Lipschitz condition and differentiability result

About lipschitz continuity of differentiable functions the following holds.

###### Theorem 1.

Let $X\mathrm{,}Y$ be Banach spaces^{} and let $A$
be a convex (see convex set), open subset of $X$.
Let $f\mathrm{:}\overline{A}\mathrm{\to}Y$ be a function which is continuous^{} in $\overline{A}$ and differentiable^{} in $A$. Then $f$ is lipschitz continuous on $\overline{A}$
if and only if the derivative^{} $D\mathit{}f$ is bounded on $A$ i.e.

$$ |

###### Proof.

Suppose that $f$ is lipschitz continuous:

$$\parallel f(x)-f(y)\parallel \le L\parallel x-y\parallel .$$ |

Then given any $x\in A$ and any $v\in X$, for all small $h\in \mathbb{R}$ we have

$$\parallel \frac{f(x+hv)-f(x)}{h}\parallel \le L.$$ |

Hence, passing to the limit $h\to 0$ it must hold $\parallel Df(x)\parallel \le L$.

On the other hand suppose that $Df$ is bounded on $A$:

$$\parallel Df(x)\parallel \le L,\forall x\in A.$$ |

Given any two points $x,y\in \overline{A}$ and given any $\alpha \in {Y}^{*}$ consider the function $G:[0,1]\to \mathbb{R}$

$$G(t)=\u27e8\alpha ,f((1-t)x+ty)\u27e9.$$ |

For $t\in (0,1)$ it holds

$${G}^{\prime}(t)=\u27e8\alpha ,Df((1-t)x+ty)[y-x]\u27e9$$ |

and hence

$$|{G}^{\prime}(t)|\le L\parallel \alpha \parallel \parallel y-x\parallel .$$ |

Applying Lagrange mean-value theorem to $G$ we know that there exists $\xi \in (0,1)$ such that

$$|\u27e8\alpha ,f(y)-f(x)\u27e9|=|G(1)-G(0)|=|{G}^{\prime}(\xi )|\le \parallel \alpha \parallel L\parallel y-x\parallel $$ |

and since this is true for all $\alpha \in {Y}^{*}$ we get

$$\parallel f(y)-f(x)\parallel \le L\parallel y-x\parallel $$ |

which is the desired claim. ∎

Title | Lipschitz condition and differentiability result |
---|---|

Canonical name | LipschitzConditionAndDifferentiabilityResult |

Date of creation | 2013-03-22 13:32:42 |

Last modified on | 2013-03-22 13:32:42 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 5 |

Author | paolini (1187) |

Entry type | Result |

Classification | msc 26A16 |