equal
deleted
inserted
replaced
824 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)" |
824 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)" |
825 by (rule continuous_on_mult [OF _ continuous_on_const]) |
825 by (rule continuous_on_mult [OF _ continuous_on_const]) |
826 |
826 |
827 lemma continuous_on_mult_const [simp]: |
827 lemma continuous_on_mult_const [simp]: |
828 fixes c::"'a::real_normed_algebra" |
828 fixes c::"'a::real_normed_algebra" |
829 shows "continuous_on s (( *) c)" |
829 shows "continuous_on s ((*) c)" |
830 by (intro continuous_on_mult_left continuous_on_id) |
830 by (intro continuous_on_mult_left continuous_on_id) |
831 |
831 |
832 lemma tendsto_divide_zero: |
832 lemma tendsto_divide_zero: |
833 fixes c :: "'a::real_normed_field" |
833 fixes c :: "'a::real_normed_field" |
834 shows "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x / c) \<longlongrightarrow> 0) F" |
834 shows "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x / c) \<longlongrightarrow> 0) F" |