If $K⊆ℝ^d$ is compact and $f:[0,T]×K×K→ℝ$ is continuous with $f(t,;⋅;,;⋅;)∈C^0+β$, is $t↦left|f(t,;⋅;,;⋅;)right|_C^0+β$ continuous?

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Let



  • $T>0$

  • $I:=(0,T]$

  • $dinmathbb N$

  • $Ksubseteqmathbb R^d$ be compact

  • $f:overline Itimes Ktimes Ktomathbb R$ be (jointly) continuous

  • $betain(0,1]$

Moreover, let $$left|gright|_C^0+beta(K):=sup_x,:y:in:Kleft|g(x,y)right|+sup_stackrelx,:y,:x',:y':in:Kx:ne:x',:y:ne:y'fracg(x,y)-g(x,y')-g(x',y)+g(x',y')rightx-x'$$ for $g:Ktimes Ktomathbb R$ and $$C^0+beta(K):=leftg:Ktimes Ktomathbb Rmidleft.$$ Assume $$f(t,;cdot;,;cdot;)in C^0+beta(K);;;textfor all tinoverline Itag1.$$




Are we able to show that $$overline Ini tmapstoleft|f(t,;cdot;,;cdot;)right|_C^0+beta(K)tag2$$ is continuous?




In general, if $E_i$ is a compact metric space and $h:E_1times E_2tomathbb R$ is (jointly) continuous, then $$E_2ni x_2mapstosup_x_1:in:X_1f(x_1,x_2)tag3$$ is continuous (the crucial argument is the uniform continuity of $h$ implied by the compactness of $E_1times E_2$).



So, the dependence on $t$ of the first term in the definition of $left|f(t,;cdot;,;cdot;)right|_C^0+beta(K)$ is continuous. What's about the second term?




analysis continuity holder-spaces




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    up vote
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    down vote

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    Let



    • $T>0$

    • $I:=(0,T]$

    • $dinmathbb N$

    • $Ksubseteqmathbb R^d$ be compact

    • $f:overline Itimes Ktimes Ktomathbb R$ be (jointly) continuous

    • $betain(0,1]$

    Moreover, let $$left|gright|_C^0+beta(K):=sup_x,:y:in:Kleft|g(x,y)right|+sup_stackrelx,:y,:x',:y':in:Kx:ne:x',:y:ne:y'fracg(x,y)-g(x,y')-g(x',y)+g(x',y')rightx-x'$$ for $g:Ktimes Ktomathbb R$ and $$C^0+beta(K):=leftg:Ktimes Ktomathbb Rmidleft.$$ Assume $$f(t,;cdot;,;cdot;)in C^0+beta(K);;;textfor all tinoverline Itag1.$$




    Are we able to show that $$overline Ini tmapstoleft|f(t,;cdot;,;cdot;)right|_C^0+beta(K)tag2$$ is continuous?




    In general, if $E_i$ is a compact metric space and $h:E_1times E_2tomathbb R$ is (jointly) continuous, then $$E_2ni x_2mapstosup_x_1:in:X_1f(x_1,x_2)tag3$$ is continuous (the crucial argument is the uniform continuity of $h$ implied by the compactness of $E_1times E_2$).



    So, the dependence on $t$ of the first term in the definition of $left|f(t,;cdot;,;cdot;)right|_C^0+beta(K)$ is continuous. What's about the second term?




    analysis continuity holder-spaces




    share|cite|improve this question























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      Let



      • $T>0$

      • $I:=(0,T]$

      • $dinmathbb N$

      • $Ksubseteqmathbb R^d$ be compact

      • $f:overline Itimes Ktimes Ktomathbb R$ be (jointly) continuous

      • $betain(0,1]$

      Moreover, let $$left|gright|_C^0+beta(K):=sup_x,:y:in:Kleft|g(x,y)right|+sup_stackrelx,:y,:x',:y':in:Kx:ne:x',:y:ne:y'fracg(x,y)-g(x,y')-g(x',y)+g(x',y')rightx-x'$$ for $g:Ktimes Ktomathbb R$ and $$C^0+beta(K):=leftg:Ktimes Ktomathbb Rmidleft.$$ Assume $$f(t,;cdot;,;cdot;)in C^0+beta(K);;;textfor all tinoverline Itag1.$$




      Are we able to show that $$overline Ini tmapstoleft|f(t,;cdot;,;cdot;)right|_C^0+beta(K)tag2$$ is continuous?




      In general, if $E_i$ is a compact metric space and $h:E_1times E_2tomathbb R$ is (jointly) continuous, then $$E_2ni x_2mapstosup_x_1:in:X_1f(x_1,x_2)tag3$$ is continuous (the crucial argument is the uniform continuity of $h$ implied by the compactness of $E_1times E_2$).



      So, the dependence on $t$ of the first term in the definition of $left|f(t,;cdot;,;cdot;)right|_C^0+beta(K)$ is continuous. What's about the second term?




      analysis continuity holder-spaces




      share|cite|improve this question













      Let



      • $T>0$

      • $I:=(0,T]$

      • $dinmathbb N$

      • $Ksubseteqmathbb R^d$ be compact

      • $f:overline Itimes Ktimes Ktomathbb R$ be (jointly) continuous

      • $betain(0,1]$

      Moreover, let $$left|gright|_C^0+beta(K):=sup_x,:y:in:Kleft|g(x,y)right|+sup_stackrelx,:y,:x',:y':in:Kx:ne:x',:y:ne:y'fracg(x,y)-g(x,y')-g(x',y)+g(x',y')rightx-x'$$ for $g:Ktimes Ktomathbb R$ and $$C^0+beta(K):=leftg:Ktimes Ktomathbb Rmidleft.$$ Assume $$f(t,;cdot;,;cdot;)in C^0+beta(K);;;textfor all tinoverline Itag1.$$




      Are we able to show that $$overline Ini tmapstoleft|f(t,;cdot;,;cdot;)right|_C^0+beta(K)tag2$$ is continuous?




      In general, if $E_i$ is a compact metric space and $h:E_1times E_2tomathbb R$ is (jointly) continuous, then $$E_2ni x_2mapstosup_x_1:in:X_1f(x_1,x_2)tag3$$ is continuous (the crucial argument is the uniform continuity of $h$ implied by the compactness of $E_1times E_2$).



      So, the dependence on $t$ of the first term in the definition of $left|f(t,;cdot;,;cdot;)right|_C^0+beta(K)$ is continuous. What's about the second term?





      analysis continuity holder-spaces





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 28 at 15:24
























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