src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
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parent 62456 11e06f5283bc
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   198 subsubsection\<open>The concept of continuously differentiable\<close>
   198 subsubsection\<open>The concept of continuously differentiable\<close>
   199 
   199 
   200 text \<open>
   200 text \<open>
   201 John Harrison writes as follows:
   201 John Harrison writes as follows:
   202 
   202 
   203 ``The usual assumption in complex analysis texts is that a path \<gamma> should be piecewise
   203 ``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise
   204 continuously differentiable, which ensures that the path integral exists at least for any continuous
   204 continuously differentiable, which ensures that the path integral exists at least for any continuous
   205 f, since all piecewise continuous functions are integrable. However, our notion of validity is
   205 f, since all piecewise continuous functions are integrable. However, our notion of validity is
   206 weaker, just piecewise differentiability... [namely] continuity plus differentiability except on a
   206 weaker, just piecewise differentiability... [namely] continuity plus differentiability except on a
   207 finite set ... [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
   207 finite set ... [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
   208 the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
   208 the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this