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Describe the graph of the function $f$ by computing some level sets and sections.

$$f:\mathbb{R}^2\to\mathbb{R},(x,y)↦ \max(|x|,|y|)$$

I have no idea how to obtain the level sets and sections of this function.

So far I have:

$c=0: x=0, y=0 \\ c=1: x=1, 0\leq y <1, \text{or }0\leq x<1,y=1\\ c=2: x=2, 0\leq y <2, \text{or }0\leq x<2,y=2\\ \text{Function not defined for }c<0.$

$c$ is the level curve value.

But I don't think this is right. How do I even graph such a function? There are infinitely many possibilities, so I'm not sure how to determine a section of a graph let alone describe the graph.

Bobby Lee
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  • Some people are fastidious about distinguish between two different arrows, "$\to$" and "$\mapsto$". The first is appropriate in the expression $f:\mathbb R^2\to\mathbb R$ and the second in $(x,y)\mapsto\max(|x|,|y|)$. ${}\qquad{}$ – Michael Hardy Feb 06 '14 at 03:55
  • You are almost there. Just draw carefuly in the $x,y$ plane what you found for $c=1,2$, guess the result for a general $c>0$, then convince yrself yr guess is correct. – Gil Bor Feb 06 '14 at 05:19
  • I'm sorry I'm not following...I have tried drawing these down and tried to interpret what they mean but I can't figure it out. – Bobby Lee Feb 06 '14 at 06:33

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When you plot the level curves for $c>0$, you should have in general the sections of $\left\{ (c,y) : 0 \leq y < c \right\} \cup \left\{ (x,c) : 0 \leq x < c \right\}$ which will have the visual appearance of the upper right quarter of a $2c \times 2c$ square, with a hole on the line $y=x$. In terms of the graph as a whole, if $(x,y)$ belongs to the part of the first quadrant that is above the $y=x$ line, then $y>x$, so $f(x,y) = y$. Similarly, for $(x,y)$ below the $y=x$ line, $x>y$ so that $f(x,y) = x$. Now, you can use the symmetry of $|x|$ and $|y|$ to extend the graph of $f$ over the whole plane.

izœc
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