16.N. Let f be a continuous function on R to R which does not take on any of its values twice. Is it true that f must either be strictly increasing or strictly decreasing?
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
exercise 16.O
Please prove EVERYTHING, (if they affirm something, please prove it even if it seems easy)
![16.N. Let f be a continuous function on R to R which does not take on any
of its values twice. Is it true that f must either be strictly inereasing or strictly
decreasing?
16.0. Let g be a function on I to R. Prove that if g takes on each of its values
exactly twice, then g cannot be continuous at every point of I.
16.P. Let f be continuous on the interval [0, 27] to R and such that f(0)
f(27). Prove that there exists a point c in this interval such that f(c) = f(c +T).
(Hint: consider g (x) = f(x) – f(x + 1).) Conclude that there are,
antipodal points on the equator of the earth which have the same temperature.
16.Q. Consider each of the functions given in Example 15.5 and either show
that the function is uniformly continuous or that it is not.
16. R. Give a proof of the Uniform Continuity Theorem 16.12 by using the
Lebesgue Covering Theorem 9.5.
at any time,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F867ba63b-b23c-4075-b706-a33f5107b31f%2Fb56b8a93-130a-49a1-9fe0-7613d8609e3d%2F4wlzy8j9_processed.png&w=3840&q=75)
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