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236 lemma frechet_derivative_const [simp]: "frechet_derivative (\<lambda>x. c) (at a) = (\<lambda>x. 0)" |
236 lemma frechet_derivative_const [simp]: "frechet_derivative (\<lambda>x. c) (at a) = (\<lambda>x. 0)" |
237 using differentiable_const frechet_derivative_works has_derivative_const has_derivative_unique by blast |
237 using differentiable_const frechet_derivative_works has_derivative_const has_derivative_unique by blast |
238 |
238 |
239 lemma frechet_derivative_id [simp]: "frechet_derivative id (at a) = id" |
239 lemma frechet_derivative_id [simp]: "frechet_derivative id (at a) = id" |
240 using differentiable_def frechet_derivative_works has_derivative_id has_derivative_unique by blast |
240 using differentiable_def frechet_derivative_works has_derivative_id has_derivative_unique by blast |
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241 |
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242 lemma frechet_derivative_ident [simp]: "frechet_derivative (\<lambda>x. x) (at a) = (\<lambda>x. x)" |
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243 by (metis eq_id_iff frechet_derivative_id) |
241 |
244 |
242 |
245 |
243 subsection \<open>Differentiability implies continuity\<close> |
246 subsection \<open>Differentiability implies continuity\<close> |
244 |
247 |
245 proposition differentiable_imp_continuous_within: |
248 proposition differentiable_imp_continuous_within: |