- 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
The set of points on the “edge” of a region
Differentiable at x = a
Elements of a matrix
See Matrix element.
Ellipsoid of revolution
A surface generated by rotating an ellipse about its major axis
Equal complex numbers
Complex numbers whose real parts are equal and whose imaginary parts are equal.
A function of the form ƒ(x) = a ? bx,where ?0, b > 0 b ?1
Higher-degree polynomial function
A polynomial function whose degree is ? 3
The complex number.
Zeros of a function that are irrational numbers.
The diagonal from the top left to the bottom right of a square matrix
Mode of a data set
The category or number that occurs most frequently in the set.
Order of an m x n matrix
The order of an m x n matrix is m x n.
Equations of the form x = ƒ(t) and y = g(t) for all t in an interval I. The variable t is the parameter and I is the parameter interval.
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.
The distance from a point on a circle (or a sphere) to the center of the circle (or the sphere).
See Reference triangle
A statistical measure that does not change much in response to outliers.
An end behavior asymptote that is a slant line
symmetric about the x-axis
A graph in which (x, -y) is on the graph whenever (x, y) is; or a graph in which (r, -?) or (-r, ?, -?) is on the graph whenever (r, ?) is
A circle with radius 1 centered at the origin.