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Math Help - epsilon delta limit proof

  1. #1
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    epsilon delta limit proof



    We have barely gone over E-D limit proofs in class this year, I kind of understand them and can do them for simple functions, but I don't know where to start on this one.
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  2. #2
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    To show that \displaystyle \lim_{\varepsilon\to0+}\frac1{V(B(\vec{x_0},\varep  silon))}\iiint_{B(\vec{x_0},\varepsilon)}f(\vec{x}  )\,dV = f(\vec{x_0}), you need to show that the difference \displaystyle \frac1{V(B(\vec{x_0},\varepsilon))}\iiint_{B(\vec{  x_0},\varepsilon)}f(\vec{x})\,dV - f(\vec{x_0}) can be made arbitrarily small, provided that \varepsilon is small enough. To get something "arbitrarily small", you usually try to make it smaller than some given \varepsilon. But we already have an \varepsilon here, so we had better choose another letter, say \eta. Let's try to make

    \displaystyle \left|\frac1{V(B(\vec{x_0},\varepsilon))}\iiint_{B  (\vec{x_0},\varepsilon)}f(\vec{x})\,dV - f(\vec{x_0})\right| < \eta, where \eta>0 is given.

    The first thing to notice is that \displaystyle \frac1{V(B(\vec{x_0},\varepsilon))}\iiint_{B(\vec{  x_0},\varepsilon)}f(\vec{x_0})\,dV = f(\vec{x_0}) (because when you integrate the constant f(\vec{x_0}) over a ball, you get that constant times the volume of the ball. Therefore

    \displaystyle \left|\frac1{V(B(\vec{x_0},\varepsilon))}\iiint_{B  (\vec{x_0},\varepsilon)}f(\vec{x})\,dV - f(\vec{x_0})\right| =  \left|\frac1{V(B(\vec{x_0},\varepsilon))}\iiint_{B  (\vec{x_0},\varepsilon)}\bigl(f(\vec{x}) - f(\vec{x_0})\bigr)\,dV\right|.

    Next, the absolute value of an integral is always less than the integral of the absolute value (of the function in the integral). Therefore

    \displaystyle \left|\frac1{V(B(\vec{x_0},\varepsilon))}\iiint_{B  (\vec{x_0},\varepsilon)}\bigl(f(\vec{x}) - f(\vec{x_0})\bigr)\,dV\right| \leqslant \frac1{V(B(\vec{x_0},\varepsilon))}\iiint_{B(\vec{  x_0},\varepsilon)}\bigl|f(\vec{x}) - f(\vec{x_0})\bigr|\,dV.

    Now, notice that if we could arrange that \bigl|f(\vec{x}) - f(\vec{x_0})\bigr| < \eta for all \vec{x}\in B(\vec{x_0},\varepsilon), then that last integral would be less than \eta, as required. But that condition, \bigl|f(\vec{x}) - f(\vec{x_0})\bigr| < \eta for all \vec{x}\in B(\vec{x_0},\varepsilon), follows from the given fact that f is continuous at \vec{x_0} (can you see why?).
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