This book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an und...This book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.
SECTION 5.1 deals with the structure of \Rn, the space of ordered n-tuples of real numbers, which we call {}. We define the sum of two vectors, the product of a vector and a real number, the...SECTION 5.1 deals with the structure of \Rn, the space of ordered n-tuples of real numbers, which we call {}. We define the sum of two vectors, the product of a vector and a real number, the length of a vector, and the inner product of two vectors. SECTION 5.2 deals with boundedness, limits, continuity, and uniform continuity of a function of n variables; that is, a function defined on a subset of \Rn.
SECTION 3.1 begins with the definition of the Riemann integral and presents the geometrical interpretation of the Riemann integral as the area under a curve. SECTION 3.5 defines the notion of a set wi...SECTION 3.1 begins with the definition of the Riemann integral and presents the geometrical interpretation of the Riemann integral as the area under a curve. SECTION 3.5 defines the notion of a set with Lebesgue measure zero, and presents a necessary and sufficient condition for a bounded function f to be Riemann integrable on an interval [a,b]; namely, that the discontinuities of f form a set with Lebesgue masure zero.
f(k)(x)=n(n−1)⋯(n−k−1)xn−k−1|x| if 1≤k≤n−1; f(n)(x)=n! if x>0; f(n)(x)=−n! if x<0; f(k)(x)=0 if k>n and x≠0; f(k)(0) does not exi...f(k)(x)=n(n−1)⋯(n−k−1)xn−k−1|x| if 1≤k≤n−1; f(n)(x)=n! if x>0; f(n)(x)=−n! if x<0; f(k)(x)=0 if k>n and x≠0; f(k)(0) does not exist if k≥n.
