- 11.3.1: Let I: JR~ JRbe defined by I(x) =x2 for x e R(a) Showthat the inver...
- 11.3.2: Let I: JR~ JRbe defined by I(x) := 1/(1 + x2) for x e R(a) Find an ...
- 11.3.3: Let 1 := [1, 00) andlet I(x) :=.JX=l for x e I. For eachE-neighborh...
- 11.3.4: Let h : JR~ JRbe definedby h(x) := 1if 0 ~ x ~ I, h(x) := 0 otherwi...
- 11.3.5: Show that if I : JR~ JRis continuous, then the set (x e JR: I(x) < ...
- 11.3.6: Showthat if I: JR~ JRis continuous,thenthe set (x e JR:I(x) ~ a} is...
- 11.3.7: Show that if I : JR~ JRis continuous, then the set (x e JR: I(x) = ...
- 11.3.8: Give an example of a function I : JR~ JRsuch that the set (x e JR: ...
- 11.3.9: Prove that I : JR~ JRis continuous if and only if for each closed s...
- 11.3.10: Let I := [a, b] and let I : I ~ JRand g : I ~ JRbe continuous funct...
Solutions for Chapter 11.3: Continuous Functions
Full solutions for Introduction to Real Analysis | 3rd Edition
A value ƒ(c) is an absolute maximum value of ƒ if ƒ(c) ? ƒ(x) for all x in the domain of ƒ.
See Cartesian coordinate system.
The function y = cot x
Distance (on a number line)
The distance between real numbers a and b, or |a - b|
The number found when the corresponding components of two vectors are multiplied and then summed
A nonnegative number that specifies how off-center the focus of a conic is
The line is a horizontal asymptote of the graph of a function ƒ if lim x:- q ƒ(x) = or lim x: q ƒ(x) = b
Instantaneous rate of change
See Derivative at x = a.
See Power function.
A graph of a polar equation of the form r2 = a2 sin 2u or r 2 = a2 cos 2u.
Linear programming problem
A method of solving certain problems involving maximizing or minimizing a function of two variables (called an objective function) subject to restrictions (called constraints)
Lower bound for real zeros
A number c is a lower bound for the set of real zeros of ƒ if ƒ(x) Z 0 whenever x < c
Multiplicative inverse of a matrix
See Inverse of a matrix
Probability of an event in a finite sample space of equally likely outcomes
The number of outcomes in the event divided by the number of outcomes in the sample space.
Quotient rule of logarithms
logb a R S b = logb R - logb S, R > 0, S > 0
The distance from a point on a circle (or a sphere) to the center of the circle (or the sphere).
Real part of a complex number
See Complex number.
Vertex of a cone
See Right circular cone.
The points x, y, 0 in Cartesian space.
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