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Thread: Continuity on the R x R plane

  1. #1
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    Continuity on the R x R plane

    Dear everyone. I have 'struggled' with the query in the attached file for a year. It would be very best if someone can enlighten me on that

    Thanks


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  2. #2
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    Re: Continuity on the R x R plane

    By the chain rule for multivariable functions, $\displaystyle h(u,v,t) = h(a+ut,b+vt)$ is a differentiable function of u, v, and t.

    The function $\displaystyle F(u,v,t) = \frac{h(a+ut,b+vt)-h(a,b)}{t}$ is the difference quotient for $\displaystyle \frac{\partial h}{\partial t}$, so its limit as t goes to 0 is the derivative, which is exactly what you specified for $\displaystyle F(u,v,0)$. So F is continuous.

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