src/HOL/Limits.thy
changeset 69064 5840724b1d71
parent 68860 f443ec10447d
child 69272 15e9ed5b28fb
equal deleted inserted replaced
69045:8c240fdeffcb 69064:5840724b1d71
   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"