Analysis: Continuous Functions

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warwick

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Jan 27, 2006
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warwick said:
For (h), I know what I wrote down is not correct. I'm not sure how to do this. Could I let the limit of f be zero and so it wouldn't matter if g is continuous or not?
http://i111.photobucket.com/albums/n149/camarolt4z28/2-3-1.jpg
Click to expand...
Consider \(\displaystyle D=[1,2]\) and two functions \(\displaystyle f(x)=0\) and \(\displaystyle g(x) = \left\{ {\begin{array}{*{20}c} {x,} & {x \in \mathbb{Q}} \\
{0,} & {x \notin \mathbb{Q}} \\
\end{array} } \right.\)

Both \(\displaystyle f~\&~f\cdot g\) are continuous but not \(\displaystyle g\).
 
pka said:
Consider \(\displaystyle D=[1,2]\) and two functions \(\displaystyle f(x)=0\) and \(\displaystyle g(x) = \left\{ {\begin{array}{*{20}c} {x,} & {x \in \mathbb{Q}} \\
{0,} & {x \notin \mathbb{Q}} \\
\end{array} } \right.\)

Both \(\displaystyle f~\&~f\cdot g\) are continuous but not \(\displaystyle g\).
Click to expand...

Ah. That's a good counterexample. I didn't even think to use that particular g(x). I should have because it's discussed in the book. Thanks.
 
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