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Hi, I have some troubles understanding the basic facts about investigating the continuity of two-variable functions.

Our professor gave us very simple example to show us the basic facts:

Very important is that projections are continuous, it means

[tex]

\pi_1 :[x,y] \longmapsto x

[/tex]

[tex]

\pi_2 :[x,y] \longmapsto y

[/tex]

are continuous.

Now let's have this function:

[tex]

f(x,y) = \sqrt{x^2+y^2}

[/tex]

It's continuous on the whole [itex]\mathbb{R}^2[/itex]. Why?

Projections are continuous

[tex]

x,y \longmapsto x^2 + y^2[/tex]...continuous

Square root is continuous on [itex][0,\infty)[/itex]

What I don't fully understand is the second step. I can't see why, from the fact that projections are continuous, we can say that

[tex]

x^2 + y^2

[/tex]

is continous. Of course I didn't expect any other result, when you look at it it's obvious that it will be continuous, but I just can't see the rigorous mathematical background.

You know, I see it is some equivalent of the limit of product of single variable function (? is it the right expression ?), but I don't think it's sufficient. I would need some equivalent for two-variable functions...

Hope you understand my problem :)

Thank you very much.

Our professor gave us very simple example to show us the basic facts:

Very important is that projections are continuous, it means

[tex]

\pi_1 :[x,y] \longmapsto x

[/tex]

[tex]

\pi_2 :[x,y] \longmapsto y

[/tex]

are continuous.

Now let's have this function:

[tex]

f(x,y) = \sqrt{x^2+y^2}

[/tex]

It's continuous on the whole [itex]\mathbb{R}^2[/itex]. Why?

**1. Step:**Projections are continuous

**2. Step:**[tex]

x,y \longmapsto x^2 + y^2[/tex]...continuous

**3. Step:**Square root is continuous on [itex][0,\infty)[/itex]

What I don't fully understand is the second step. I can't see why, from the fact that projections are continuous, we can say that

[tex]

x^2 + y^2

[/tex]

is continous. Of course I didn't expect any other result, when you look at it it's obvious that it will be continuous, but I just can't see the rigorous mathematical background.

You know, I see it is some equivalent of the limit of product of single variable function (? is it the right expression ?), but I don't think it's sufficient. I would need some equivalent for two-variable functions...

Hope you understand my problem :)

Thank you very much.

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