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02 intégrale sur un segment d une fonction continue par morceaux

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02 intégrale sur un segment d une fonction continue par morceaux

  1. 1. ! " # ! < $ % & ! ' & ! [ ]# $#"""#( ) * =<<<= """) " % $#"""#( )=σ $+#"""+#+(+ )=σ & ! [ ]# # * +σ * σ * { } { }+#"""+#+#"""# )) ⊂ % $#"""#( )=σ $+#"""+#+(+ )=σ & ! * * [ ]# # * , & * & ! * +σ * σ & { } { }+#"""+#+#"""# )) ∪ " -$ . ' * [ ]# [ ]# & ! $#"""#( )=σ [ ]# * * ! ! ] [#− # " / & ! σ & " ' ! * # σ & ! & # & ! * σ & " [ ]# * & ! # & " 0 [ ]# # & ! & ! σ +σ # & ! ++σ * σ +σ & " * & # * # [ ]# # & ! & ++σ " 1 # * & [ ]# (* [ ]# # $ 2& 3 2& [ ]# ( ! $" touscours.net
  2. 2. $ % [ ]# # $#"""#( )=σ & ! & " 4 # [ ]# # ! ! ! ] [#− " ! & = −−= $($#( σ & ! σ & " ' & * +σ σ , # $#($+#( σσ = " 1 # & , * +σ * σ # $#($+#( σσ = " 0 # # & ! ++σ * +σ σ # $#($++#($+#( σσσ == " ' [ ]# ! $#( σ # & ! σ & " [ ]# " # ! [ ] = −−= # $( " ! 5 * 6 & * & " ' * * [ ]# $( − 7 ! " ' * ! ( [ ]# # λ µ $ 3 ! )≥ [ ]# # [ ] ) # ≥ 3 / [ ] [ ] [ ] +=+ ### µλµλ 3 ! % << # [ ] [ ] [ ] += ### ( ! * & [ ]# [ ]# $" 3 % ≤ [ ]# # [ ] [ ] ≤ ## ( ! $ % [ ]# # * & " ' " % $(− ε & ϕ [ ]# * ( + 3 3 * ≤ϕ $ touscours.net
  3. 3. 8 % $(+ ε & ψ [ ]# * ( + 3 3 * ψ≤ $ / & $(− ε $(+ ε ! $(− ε # $(+ ε " % $(− & $(− ε " % $(+ & $(+ ε " / & $(− $(+ & ! # $(− $(+ # ϕ ψ [ ]# * ψϕ ≤≤ # ψϕ ≤ # [ ] [ ] ≤ ## ψϕ " 1 $(− & # $(− # $(+ & # $(+ " # $($( +− ≤ " % & # * & [ ]# # [ ]# ! & " 1 # & [ ]# # * & [ ]# " & & 9 " ' * # # & & [ ]# # * : ! ( # ! * # * # $(− ε $(+ ε ; $ 0 # & [ ]# # [ ]# [ ]# # $( ( 2 # ! & # * $" ' ! * & ! 5 * 6 & * & " $ 1 % [ ]# " ' * [ ]# & ! $#"""#( )=σ [ ]# * * # ! ! ] [#− # − " / & ! σ & " ' ! * [ ]# # & ! σ & # & ! * σ & " * [ ]# * & (! ; $# * touscours.net
  4. 4. < ( $" % [ ]# # $#"""#( ) & ! & " # * [ ]# # ! ! ] [#− & [ ]#− " * [ ]# & # ! # * [ ]# # [ ]#− # & " & # & [ ]# # , [ ] [ ] [ ]− ### ) ) #"""##$(#"""$(#$( ' # * & [ ]# [ ]# " 1 # * [ ]# 2& 3 2& [ ]# " -$ 0 2 % [ ]# " # ε # ϕ [ ]# * εϕ ≤− " 1 • = 7 [ ]# " % )>ε [ ]# ( 2 > $# )>α * [ ] [ ] ( )εα <−<−∈∀∈∀ $+($(+##+## ' 2 * α< − # & ! 2 $#"""#( )=σ [ ]# [ ] +=∈∀ ##) # ! − = ( & ! 2 σ $ ' 2 ϕ [ ] [ [ $($(#### −− =∈∀∈∀ ϕ $($( =ϕ " εϕ ≤− % [ ]#∈ 3 % = # εϕ ≤=− )$($( 3 % # [ ]#∈ * [ [#−∈ " touscours.net
  5. 5. ? εϕ <−=− − $($($($( # 2 ! * # [ [#−∈ # α<−≤− −− • % * ' & ! $#"""#( )=σ & # # * [ ]# # 2 $# * [ ]#− * : ! ] [#− " 0 )>ε # * * # * [ ]#− ϕ * [ ] εϕ ≤−∈∀ − $($(## " ' ϕ [ ]# [ ] $($(##) =∈∀ ϕ # [ ] ] [ $($(#### ϕϕ =∈∀∈∀ − " ϕ ! [ ]# # εϕ ≤− " 2 # 2 (! $ % [ ]# " # ε # ϕ ψ [ ]# * ψϕ ≤≤ εϕψ ≤− / ! @ # ϕ * εϕ ≤− # εϕϕ −=+ εϕψ +=+ ++ ψϕ ≤≤ εϕψ ≤− # ! # ψϕ ≤≤ εϕψ ≤− # ! εϕ ≤− " $ * & 2 [ ]# & [ ]# " 1 % [ ]# " % )>ε " % 2 # ϕ ψ [ ]# * ψϕ ≤≤ εϕψ ≤− " # # $(− ∈εϕ $(+ ∈εψ # [ ] [ ] ≤≤≤ +− ## $($( ψϕ " # [ ] [ ] [ ] [ ] εεψϕψϕ $( #### −=≤−=− 1 ε$($($() −≤−≤ −+ ! & * * )>ε # # # * )$($( =− −+ # # & [ ]# " touscours.net
  6. 6. ' * [ ]# * # ε # ϕ ψ [ ]# * ψϕ ≤≤ εϕψ ≤− " & # * & " 0 # * * 2 5 [ ]# 6# # 2 # ! * 2 5 [ ]# & 6" ' & & " A * • 0 & & " % * [ ]#) ( + 3 3 $( = ∈ # ) $ % ϕ [ ]#) ≤ϕ # ! ] [#− & ! & ϕ # ! ϕ ) * ! ) ] [#− (* $" 1 )$( =− " 1 @ # $( =+ # 7 & ( 9 $" • 0 # @ # & % [ ]#) $( = )≠ # ) " 1 ,# * )" 1 # ! ϕ ψ [ ]#) * ψϕ ≤≤ ≤−ϕψ " 0 # ! 3 ! [ ]α#) ! ) ≤< α # ϕ ψ ! [ ]#) & ! & ϕ ψ # ! " # ε * ) << ε " [ ]#ε # & # ! # ϕ ψ [ ]#ε # ε " % ϕ ψ [ ]#) $( −=ϕ $( =ψ [ [ε#)∈ # * ϕ ψ [ ]#) # * ψϕ ≤≤ # * [ ] [ ] [ ] [ ] εεεεϕψϕψϕψ εε 8$( ##)#)#) ≤−+≤−+−=− 1 7# 2 )$($( =− −+ " • 0 # * " % [ ]#) [ ] $( = )≠ # ) " # ( { })B∈ $" # * ε 2 (! * C ( * ε< $# [ ]#) ) # ! ] ]# $ # # & " ( −= π $" touscours.net

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