Lipschitz continuous : Lipschitz maps
In mathematics, a function f : M → N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) ≤ K d(x, y) for all x and y in M. In this case, K is called the Lipschitz constant of the map. The name comes from the German mathematician Rudolf Lipschitz[?].Every Lipschitz continuous map is uniformly continuous and hence continuous.
Lipschitz continuous maps with Lipschitz contant K < 1 are called contractions; they are the subject of the Banach fixed point theorem.
Lipschitz continuity is an important condition in the existence and uniqueness theorem for ordinary differential equations.
If U is a subset of the metric space M and f : U → R is a real-valued Lipschitz continuous map, then there always exist Lipschitz continuous maps M → R which extend f and have the same Lipschitz constant as f.
A Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure zero). If K is the Lipschitz constant of f, then |f'(x)| ≤ K whenever the derivative exists. Conversely, if f : I → R is a differentiable map with bounded derivative, |f'(x)| ≤ L for all x in I, then f is Lipschitz continuous with Lipschitz constant K ≤ L, a consequence of the mean value theorem.
This system of doctrine I have termed "mysticism"--a word which is
religious and philosophical.html">philosophical thought, from the noblest to the most degraded.
emphasises the unity between Creator and creature, though not
theology asserts, has sprung from God, but has fallen away from Him
whereby, if he will follow the narrow road of self-renunciation, he may
and ultimately indissolubly united to God in love. God is at once the
End of all existence; and He is also the Way to that End. In Christian
CHRIST also is the means towards the attainment of this end.
By mystical philosophy I mean that system of philosophical thought
the spiritual may be perceived immanent in the things of this world,
As one of the _Golden Verses_ attributed to PYTHAGORAS, which I have
is in all things alike"; commenting upon which, HIEROCLES, writing in
Universe after the Divine Measure and Proportion, made it in all things
Of all the different species, diffused throughout the whole, it made,
to the copy the perfections of the Original."[1] We have, however,
need be said here concerning it.
(trans. by N. ROWE, 1906), pp. 101 and 102.
as the attempt to realise the presence of the living God in the soul
in thought and feeling, the immanence of the temporal in the eternal,
[2] WILLIAM RALPH INGE, M.A.: _Christian Mysticism_ (the Bampton Lectures,
Now, doctrines such as these were not only very prevalent during
great antiquity, and were undoubtedly believed in by the learned
as some think, alchemy originated, though the evidence,
and post-Christian origin for the central theorem of alchemy.
.
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