equal
deleted
inserted
replaced
270 \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))" |
270 \<bar>(f(v) - f(u)) - (f'(x) * (v - u))\<bar> \<le> e * (v - u))" |
271 proof (clarsimp) |
271 proof (clarsimp) |
272 fix x :: real assume "a \<le> x" and "x \<le> b" |
272 fix x :: real assume "a \<le> x" and "x \<le> b" |
273 with f' have "DERIV f x :> f'(x)" by simp |
273 with f' have "DERIV f x :> f'(x)" by simp |
274 then have "\<forall>r>0. \<exists>s>0. \<forall>z. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < r" |
274 then have "\<forall>r>0. \<exists>s>0. \<forall>z. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < r" |
275 by (simp add: DERIV_iff2 LIM_def) |
275 by (simp add: DERIV_iff2 LIM_eq) |
276 with `0 < e` obtain s |
276 with `0 < e` obtain s |
277 where "\<forall>z. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < e/2" and "0 < s" |
277 where "\<forall>z. z \<noteq> x \<and> \<bar>z - x\<bar> < s \<longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < e/2" and "0 < s" |
278 by (drule_tac x="e/2" in spec, auto) |
278 by (drule_tac x="e/2" in spec, auto) |
279 then have strad [rule_format]: |
279 then have strad [rule_format]: |
280 "\<forall>z. \<bar>z - x\<bar> < s --> \<bar>f z - f x - f' x * (z - x)\<bar> \<le> e/2 * \<bar>z - x\<bar>" |
280 "\<forall>z. \<bar>z - x\<bar> < s --> \<bar>f z - f x - f' x * (z - x)\<bar> \<le> e/2 * \<bar>z - x\<bar>" |