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$$ In this case we say that $f$ is differentiable at $c$, and we write $f^\prime(c)$ for $L$. In other words, the derivative of $f$ at $c$ is given by the limit $$ \tag{2} f^\prime(c) = \lim_{x \to c} \frac{ f(x) - f(c)}{ x-c} $$ provided this limit exists. (We allow the possibility that $c$ may be the endpoint of the interval. )
\tag{2} $$ And for $x=c$ both sides of (2) equal $0$. Therefore we can conclude that $$ \left\lvert f(x)-f(c) - (x-c)f^\prime(c) \right\rvert \leq \varepsilon \lvert x - c \rvert \tag{2'} $$ for all $x \in I$ for which $c-\delta(\varepsilon) < x < c+\delta(\varepsilon)$.
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, Donald R. Sherbert In recent years, mathematics has become valuable in many areas, including economics and management science as well as the physical sciences, engineering and computer science. Therefore, this book provides the fundamental concepts and techniques of real analysis for readers in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations and extend ideas to a new context. Like the first two editions, this edition maintains the same spirit and user-friendly approach with some streamlined arguments, a few new examples, rearranged topics, and a new chapter on the Generalized Riemann Integral.
GitHub - ChrisHayduk/The-Elements-of-Real-Analysis-Solutions: Solutions to problems from "The Elements of Real Analysis" by Robert G. Bartle, Second Edition Solutions to problems from "The Elements of Real Analysis" by Robert G. Bartle, Second Edition Files Permalink Failed to load latest commit information. Type Name Latest commit message Commit time About You can't perform that action at this time.