Partial derivative of a composite function - weak conditions

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Let $h=h(u,w):mathbbRtimes [0,+infty)longmapsto mathbbR$ be a function such that $h,h_uin C(mathbbRtimes [0,+infty))$.



Consider the function $$f=f(x,t,s):mathbbRtimes [0,+infty)times [0,+infty)longmapsto mathbbR\ ;;;;;;;;;;;; (x,t,s) longrightarrow h(x-c(t-s),s).$$



Is it true that partial derivatives of $f$ with respect to $x$ and $t$, $fracpartial fpartial x$ and $fracpartial fpartial t$, exist and are continuous on $mathbbRtimes [0,+infty)times [0,+infty)$?



I know that if $h$ is supposed to be $C^1(mathbbRtimes [0,+infty))$ then we can consider $$psi=psi(x,t,s):mathbbRtimes [0,+infty)times [0,+infty)longmapsto mathbbR^2\ ;;;;;;;;;;;; (x,t,s) longrightarrow (x-c(t-s),s)$$wich is clearly $C^1(mathbbRtimes [0,+infty)times [0,+infty))$.



Then the composition $$f=hcircpsi in C^1(mathbbRtimes [0,+infty)times [0,+infty)).$$
In particular, $fracpartial fpartial x$ and $fracpartial fpartial t$ exist and are continuous on $C^1(mathbbRtimes [0,+infty)times [0,+infty))$ and by chain rule $$fracpartial fpartial x=fracpartial hpartial ucdotfracpartial (x-c(t-s))partial x+fracpartial hpartial wcdotfracpartial spartial x=h_u(x-c(t-s),s),$$$$fracpartial fpartial t=fracpartial hpartial ucdotfracpartial (x-c(t-s))partial t+fracpartial hpartial wcdotfracpartial spartial t=(-c)cdot h_u(x-c(t-s),s).$$



Can I use the chain rule even if it is just $h,h_uin C(mathbbRtimes [0,+infty))$ and conclude that both $fracpartial fpartial x$ and $fracpartial fpartial t$ are continuous on $mathbbRtimes [0,+infty)times [0,+infty)$?



Thanks in advance!!




real-analysis multivariable-calculus partial-derivative function-and-relation-composition chain-rule




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    Let $h=h(u,w):mathbbRtimes [0,+infty)longmapsto mathbbR$ be a function such that $h,h_uin C(mathbbRtimes [0,+infty))$.



    Consider the function $$f=f(x,t,s):mathbbRtimes [0,+infty)times [0,+infty)longmapsto mathbbR\ ;;;;;;;;;;;; (x,t,s) longrightarrow h(x-c(t-s),s).$$



    Is it true that partial derivatives of $f$ with respect to $x$ and $t$, $fracpartial fpartial x$ and $fracpartial fpartial t$, exist and are continuous on $mathbbRtimes [0,+infty)times [0,+infty)$?



    I know that if $h$ is supposed to be $C^1(mathbbRtimes [0,+infty))$ then we can consider $$psi=psi(x,t,s):mathbbRtimes [0,+infty)times [0,+infty)longmapsto mathbbR^2\ ;;;;;;;;;;;; (x,t,s) longrightarrow (x-c(t-s),s)$$wich is clearly $C^1(mathbbRtimes [0,+infty)times [0,+infty))$.



    Then the composition $$f=hcircpsi in C^1(mathbbRtimes [0,+infty)times [0,+infty)).$$
    In particular, $fracpartial fpartial x$ and $fracpartial fpartial t$ exist and are continuous on $C^1(mathbbRtimes [0,+infty)times [0,+infty))$ and by chain rule $$fracpartial fpartial x=fracpartial hpartial ucdotfracpartial (x-c(t-s))partial x+fracpartial hpartial wcdotfracpartial spartial x=h_u(x-c(t-s),s),$$$$fracpartial fpartial t=fracpartial hpartial ucdotfracpartial (x-c(t-s))partial t+fracpartial hpartial wcdotfracpartial spartial t=(-c)cdot h_u(x-c(t-s),s).$$



    Can I use the chain rule even if it is just $h,h_uin C(mathbbRtimes [0,+infty))$ and conclude that both $fracpartial fpartial x$ and $fracpartial fpartial t$ are continuous on $mathbbRtimes [0,+infty)times [0,+infty)$?



    Thanks in advance!!




    real-analysis multivariable-calculus partial-derivative function-and-relation-composition chain-rule




    share|cite|improve this question





















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      Let $h=h(u,w):mathbbRtimes [0,+infty)longmapsto mathbbR$ be a function such that $h,h_uin C(mathbbRtimes [0,+infty))$.



      Consider the function $$f=f(x,t,s):mathbbRtimes [0,+infty)times [0,+infty)longmapsto mathbbR\ ;;;;;;;;;;;; (x,t,s) longrightarrow h(x-c(t-s),s).$$



      Is it true that partial derivatives of $f$ with respect to $x$ and $t$, $fracpartial fpartial x$ and $fracpartial fpartial t$, exist and are continuous on $mathbbRtimes [0,+infty)times [0,+infty)$?



      I know that if $h$ is supposed to be $C^1(mathbbRtimes [0,+infty))$ then we can consider $$psi=psi(x,t,s):mathbbRtimes [0,+infty)times [0,+infty)longmapsto mathbbR^2\ ;;;;;;;;;;;; (x,t,s) longrightarrow (x-c(t-s),s)$$wich is clearly $C^1(mathbbRtimes [0,+infty)times [0,+infty))$.



      Then the composition $$f=hcircpsi in C^1(mathbbRtimes [0,+infty)times [0,+infty)).$$
      In particular, $fracpartial fpartial x$ and $fracpartial fpartial t$ exist and are continuous on $C^1(mathbbRtimes [0,+infty)times [0,+infty))$ and by chain rule $$fracpartial fpartial x=fracpartial hpartial ucdotfracpartial (x-c(t-s))partial x+fracpartial hpartial wcdotfracpartial spartial x=h_u(x-c(t-s),s),$$$$fracpartial fpartial t=fracpartial hpartial ucdotfracpartial (x-c(t-s))partial t+fracpartial hpartial wcdotfracpartial spartial t=(-c)cdot h_u(x-c(t-s),s).$$



      Can I use the chain rule even if it is just $h,h_uin C(mathbbRtimes [0,+infty))$ and conclude that both $fracpartial fpartial x$ and $fracpartial fpartial t$ are continuous on $mathbbRtimes [0,+infty)times [0,+infty)$?



      Thanks in advance!!




      real-analysis multivariable-calculus partial-derivative function-and-relation-composition chain-rule




      share|cite|improve this question











      Let $h=h(u,w):mathbbRtimes [0,+infty)longmapsto mathbbR$ be a function such that $h,h_uin C(mathbbRtimes [0,+infty))$.



      Consider the function $$f=f(x,t,s):mathbbRtimes [0,+infty)times [0,+infty)longmapsto mathbbR\ ;;;;;;;;;;;; (x,t,s) longrightarrow h(x-c(t-s),s).$$



      Is it true that partial derivatives of $f$ with respect to $x$ and $t$, $fracpartial fpartial x$ and $fracpartial fpartial t$, exist and are continuous on $mathbbRtimes [0,+infty)times [0,+infty)$?



      I know that if $h$ is supposed to be $C^1(mathbbRtimes [0,+infty))$ then we can consider $$psi=psi(x,t,s):mathbbRtimes [0,+infty)times [0,+infty)longmapsto mathbbR^2\ ;;;;;;;;;;;; (x,t,s) longrightarrow (x-c(t-s),s)$$wich is clearly $C^1(mathbbRtimes [0,+infty)times [0,+infty))$.



      Then the composition $$f=hcircpsi in C^1(mathbbRtimes [0,+infty)times [0,+infty)).$$
      In particular, $fracpartial fpartial x$ and $fracpartial fpartial t$ exist and are continuous on $C^1(mathbbRtimes [0,+infty)times [0,+infty))$ and by chain rule $$fracpartial fpartial x=fracpartial hpartial ucdotfracpartial (x-c(t-s))partial x+fracpartial hpartial wcdotfracpartial spartial x=h_u(x-c(t-s),s),$$$$fracpartial fpartial t=fracpartial hpartial ucdotfracpartial (x-c(t-s))partial t+fracpartial hpartial wcdotfracpartial spartial t=(-c)cdot h_u(x-c(t-s),s).$$



      Can I use the chain rule even if it is just $h,h_uin C(mathbbRtimes [0,+infty))$ and conclude that both $fracpartial fpartial x$ and $fracpartial fpartial t$ are continuous on $mathbbRtimes [0,+infty)times [0,+infty)$?



      Thanks in advance!!





      real-analysis multivariable-calculus partial-derivative function-and-relation-composition chain-rule





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      asked Jul 21 at 10:58









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