prove or disprove Lipchitz function

Question

Prove or disprove that $f(x)= -7x-3$ is Lipchitz on $\mathbb{R}$

My answer is:

$|f(x)-f(y)|=|-7x-3-(-7y-3)|$

$=|-7x+7y|=|-7(x-y)|=7|x-y|$

If the answer is equal not less than or equal can we say it is Lipchitz? And also they asked if it is Lipchitz in $\mathbb{R}$ so is this an answer on $\mathbb{R}$?

yes true so if my answer is = so true and same if my answer is <. — Lama Bitar, Jan 09 '21 at 14:02
but in class they teach us lipchitz over close interval but here its is over R is it te same?; — Lama Bitar, Jan 09 '21 at 14:03
You can continue with $\ldots =7|x-y| \leq 7|x-y|$ OR $\ldots =7|x-y| < 7.1|x-y|$ OR $\ldots =7|x-y| \leq 7.1|x-y|$ OR $\ldots =7|x-y| \leq 42|x-y|$ etc. This last one is because for something like, this one likes to introduce the number 42! — Dave L. Renfro, Jan 09 '21 at 14:03
I think there is a theorem that if $f$ differentiable and $f'$ is bounded then it is lipschitz. Here $f'=-7$ for all $x$. So it satisfies Lipschitz contion on $\mathbb R$. Please let me know if there are no such theorems. — PNDas, Jan 09 '21 at 14:05
Please let me know if there are no such theorems. --- If true, this would be VERY difficult to know (and even more difficult to show)! Fortunately, this is NOT true --- fortunately, because a "disproof that there are no such theorems" can be done by giving a counterexample. — Dave L. Renfro, Jan 09 '21 at 14:12

score 1 · Accepted Answer · answered Jan 09 '21 at 14:15

Your proof is fine. For your query in comment:

The definition of Lipschitz function is:

If $A\subseteq\mathbb{R}$ and $f:A→\mathbb{R}$ is function. Then, $f$ is said to be Lipschitz on $A$ if there exists a constant $K>0$ such that,

$|f(x)-f(y)|≤K|x-y|$ for all $x,y\in A$

See that, in definition $A$ is subset of $\mathbb{R}$. So $A$ may be equals to $\mathbb{R}$ itself.

prove or disprove Lipchitz function

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