Transcription of Chapter 4
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Chapter 4
Chapter4 Differentiable FunctionsA differentiable function is a function that can be approximated locally by a The derivativeDe nition thatf: (a, b) Randa < c < b. Thenfis differentiableatcwith derivativef (c) iflimh 0[f(c+h) f(c)h]=f (c).The domain off is the set of pointsc (a, b) for which this limit exists. If thelimit exists for everyc (a, b) then we say thatfis differentiable on (a, b).Graphically, this definition says that the derivative offatcis the slope of thetangent line toy=f(x) atc, which is the limit ash 0 of the slopes of the linesthrough (c, f(c)) and (c+h, f(c+h)).We can also writef (c) = limx c[f(x) f(c)x c],since ifx=c+h, the conditions 0<|x c|< and 0<|h|< in the definitionsof the limits are equivalent. The ratiof(x) f(c)x cis undefined (0/0) atx=c, but it doesn t have to be defined in order for the limitasx cto continuity, differentiability is a local property.
40 4. Differentiable Functions where A ⊂ R, then we can define the differentiability of f at any interior point c ∈ A since there is an open interval (a,b) ⊂ A with c ∈ (a,b). 4.1.1. Examples of derivatives. Let us give a number of examples that illus-trate differentiable and non-differentiable functions.
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