202 unfolding continuous_def by (rule tendsto_snd) |
202 unfolding continuous_def by (rule tendsto_snd) |
203 |
203 |
204 lemma continuous_Pair[continuous_intros]: "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))" |
204 lemma continuous_Pair[continuous_intros]: "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))" |
205 unfolding continuous_def by (rule tendsto_Pair) |
205 unfolding continuous_def by (rule tendsto_Pair) |
206 |
206 |
207 lemma continuous_on_fst[continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. fst (f x))" |
207 lemma continuous_on_fst[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. fst (f x))" |
208 unfolding continuous_on_def by (auto intro: tendsto_fst) |
208 unfolding continuous_on_def by (auto intro: tendsto_fst) |
209 |
209 |
210 lemma continuous_on_snd[continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. snd (f x))" |
210 lemma continuous_on_snd[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. snd (f x))" |
211 unfolding continuous_on_def by (auto intro: tendsto_snd) |
211 unfolding continuous_on_def by (auto intro: tendsto_snd) |
212 |
212 |
213 lemma continuous_on_Pair[continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. (f x, g x))" |
213 lemma continuous_on_Pair[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. (f x, g x))" |
214 unfolding continuous_on_def by (auto intro: tendsto_Pair) |
214 unfolding continuous_on_def by (auto intro: tendsto_Pair) |
215 |
215 |
216 lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a" |
216 lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a" |
217 by (fact continuous_fst) |
217 by (fact continuous_fst) |
218 |
218 |