- published: 12 Jan 2009
- views: 188494
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
In calculus, a function defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely increasing or decreasing. It is called monotonically increasing (also increasing or nondecreasing), if for all
and
such that
one has
, so
preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or nonincreasing) if, whenever
, then
, so it reverses the order (see Figure 2).
If the order in the definition of monotonicity is replaced by the strict order
, then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. Functions that are strictly increasing or decreasing are one-to-one (because for
not equal to
, either
or
and so, by monotonicity, either
or
, thus
is not equal to
.)
Increasing/Decreasing , Local Maximums/Minimums
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Increasing/Decreasing , Local Maximums/Minimums - The basic idea! In this video, I just give a graph and discuss intervals where the function is increasing and decreasing. I also discuss local maximums and minimums.
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In all the examples I have used in this video, by increasing or decreasing I mean strictly increasing or strictly decreasing unless otherwise stated.
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Fuck the police fuck all the schools
Fuck all the people that wan' live in woods
Fuck all the fisher and fuckin fish sellers
But nobody wants to giv'it and tell her
Fuck politicians and all the laws done
Fuck all the people that don't wanna come
A don't think so to you're face's still blue
At least I don't care so FUCK OFF YOU TOO...
so Fuck off you too...
Step off step off step off from me
so Fuck off you too... [x2]
Fuck all the soldiers Fuck all the wars
Fuck politicians again fuck all the world
Fuck all the things that I don't like seize the moment shut up an die
Tell me if you think that I should go away tell me first why and show me the way
I don't think so too you're face's still blue
At least I don't care so FUCK OFF YOU TOO...
so Fuck off you too...
Step off step off step off from me
so Fuck off you too... [x2]
Make me happy one last time without a word just step off
And I'll warn you one more time cause there is nothing I can take
so Fuck off you too...
Step off step off step off from me