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I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...
I am focused on Chapter 6: Differentiation ...
I need help in fully understanding the corollary to Theorem 6.2.1 ...
Theorem 6.2.1 and its corollary ... ... read as follows:
I am trying to fully understand the proof of the corollary ...
I was given the following proof by GJA (Math Help Boards) ... ...
"Either the derivative of ##f## at ##x=c## exists or it doesn't, and these are the only two possibilities. If it does, then ##f′(c)=0## from the theorem."BUT ... GJA's proof does not use the Corollary's assumption of continuity of ##f## ...
Is something amiss with GJA's proof ... ?
Peter*** EDIT ***
Note that Manfred Stoll in his book "Introduction to Real Analysis" gives the same theorem and corollary (Theorem 5.2.2 and Corollary 5.2.3) and again gives the condition that ##f## is continuous ... in Stoll's case that f is continuous on ##[a, b]## ...
I am focused on Chapter 6: Differentiation ...
I need help in fully understanding the corollary to Theorem 6.2.1 ...
Theorem 6.2.1 and its corollary ... ... read as follows:
I was given the following proof by GJA (Math Help Boards) ... ...
"Either the derivative of ##f## at ##x=c## exists or it doesn't, and these are the only two possibilities. If it does, then ##f′(c)=0## from the theorem."BUT ... GJA's proof does not use the Corollary's assumption of continuity of ##f## ...
Is something amiss with GJA's proof ... ?
Peter*** EDIT ***
Note that Manfred Stoll in his book "Introduction to Real Analysis" gives the same theorem and corollary (Theorem 5.2.2 and Corollary 5.2.3) and again gives the condition that ##f## is continuous ... in Stoll's case that f is continuous on ##[a, b]## ...