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I have a question.

I have to check whether $0\in{\bf R}^2$ is a interior point or a boundary point or neither. But I don't exactly know what the difference is, could someone explain me that?

And maybe can you help with the following example. $W=\{\,x\in{\bf R}^2:|x_1|\le2,|x_2|\le1\,\}$ Thank you

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    If you could give us the precise definition of interior/boundary points you are using here, that'd be helpful – 2017-01-29

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Interior to what set ? $\mathbb{R}^2$? For the second part of the question. Then clearly $0$ is an interior point of $W$ since you can take the open ball $B$ of radius $1/2$ centered in $0$ and $B\subsetneq W$.

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    How did you get a radius of a 1/2? – 2017-01-29
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    The idea is that $W$ is a rectangle of sides 4x2 with lowest corner to the left at coordinates $(-2,-1)$ and upper rightmost corner at $(2,1)$ so an open ball of radius $1/2$ and centered in $(0,0)$ perfectly fits in $W$. Of course when you write $0\in\mathbb{R}^2$ is suppose you mean $(0,0)\in\mathbb{R}^2$. – 2017-01-29
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    Thank you, but how do you get a radius 0.5? If I use the formula, I get sqrt((x-xo)^2+(y-y0)^2) – 2017-01-29
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    And that is sqrt (x1^2+x2^2) – 2017-01-29
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    Yes, indeed the point $(-2,-1)$ is outside the open ball we were taliking about. Same for $(2,1)$... – 2017-01-29