Proving continuity of f(x,y) = g(x)p(y)

  • Thread starter Castilla
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  • #1
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I know this must be easy, but...

Say real functions g(x) and p(y) are continuous and f(x,y) = g(x)p(y). How to proof rigorously the continuity of f in a point (x1,y1)?

In other words, how to obtain l g(x)p(y) - g(x1)p(y1) l < epsilon (for any epsilon).

I can prove that l g(x)p(y1) - g(x1)p(y) l < any epsilon, but I cant see how to go from here to there. I am trying all variations of the triangular inequality, to no avail.

Thanks for your help.
 

Answers and Replies

  • #2
31
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We want to prove
[tex]\forall \varepsilon>0\quad\exists\delta>0\quad\forall (x,y)\in B_{\delta}(x_0,y_0):f(x,y)\in B_{\varepsilon}(f(x_0,y_0))[/tex]

Let fix epsilon and we want to find such delta. From definition of ball B, we compute

[tex]\begin{array}{ll}|f(x,y)-f(x_0,y_0)|:&=|g(x)p(y)-g(x_0)p(y_0)|\\&=|g(x)p(y)-g(x)p(y_0)+g(x)p(y_0)-g(x_0)p(y_0)|\\&\le|g(x)p(y)-g(x)p(y_0)|+|g(x)p(y_0)-g(x_0)p(y_0)|\\&=|p(y)-p(y_0)||g(x)|+|g(x)-g(x_0)||p(y_0)|\end{array}[/tex]

Now, I think you are able to complete this proof. Use the continuity of g and p.
 
  • #3
22,089
3,297
Hi Castilla! :smile:

[tex]|g(x)p(y)-g(x_1)p(y_1)|=|(g(x)p(y)-g(x_1)p(y))+(g(x_1)p(y)-g(x_1)p(y_1))|\leq |g(x)-g(x_1)||p(y)|+|g(x_1)||p(y)-p(y_1)|[/tex]

I'll let you continue from there.
 
  • #4
240
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Thank you, stanley and micromass. You enlightened my mind.
 
  • #5
240
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Just one question. Looking at both answers, basically the same, it seems that I need that the functions g and p to be bounded.

So I would need not only that g and p are continuous, but also that they're continuous in a compact set?

thanks again!
 
  • #6
disregardthat
Science Advisor
1,861
34
You don't need that, but good catch!

You have

|g(x)p(y)-g(x_0)p(y_0)| = |(g(x)-g(x_0))(p(y)-p(y_0)) +p(y_0)(g(x)-g(x_0)) + g(x_0)(p(y)-p(y_0))| <= |g(x)-g(x_0)||p(y)-p(y_0)| +|p(y_0)||g(x)-g(x_0)| + |g(x_0)|p(y)-p(y_0)|
 
Last edited:
  • #7
240
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Excellent explanation. Thank you.
 
  • #8
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
You could have used the fact multiplication is continuous....
 

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