# Complex exponential has \$1\$ as Lipschitz constant.

(In the following, Lipschitz constant does not mean “best Lipschitz constant”.)

I’ve just read this in a book that I highly regard:

Moreover, by mean value theorem, \$uto e^{iu}\$ is Lipschitz continuous with Lipschitz constant \$1\$.

How does the author infer this from the mean value theorem ? The theorem applies only for real-valued functions.

The mean value theorem shows nontheless that \$sin\$ and \$cos\$ have Lipschitz constant \$1\$, and therefore \$uto e^{iu}\$ has Lipschitz constant \$2\$.

How can this be lowered to \$1\$ ?