continuity

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April 9th 2010, 12:28 PM
sfspitfire23

continuity

If $f:I\rightarrow R$ and $g: \rightarrow R$ are any functions defined on $I$ , then we can form new functions $max\{f,g\}(x)=max\{f(x),g(x)\}$ for all $x\in I$ , and similarly for $min\{f,g\}$

I'm asked to sketch a graph of $max\{f,g\}$ and $min\{f,g\}$ if $f(x)=sin x$ and $g(x)=cos x$ on $I=[-2\pi , 2\pi ]$

What do the max/min thing mean? Basically, what is the first paragraph telling me? Thanks for any help.
April 9th 2010, 12:33 PM
Drexel28

Quote:

Originally Posted by sfspitfire23

If $f:I\rightarrow R$ and $g: \rightarrow R$ are any functions defined on $I$ , then we can form new functions $max\{f,g\}(x)=max\{f(x),g(x)\}$ for all $x\in I$ , and similarly for $min\{f,g\}$

I'm asked to sketch a graph of $max\{f,g\}$ and $min\{f,g}$ if $f(x)=sin x$ and $g(x)=cos x$ on $I=[-2\pi , 2\pi ]$

What do the max/min thing mean? Basically, what is the first paragraph telling me? Thanks for any help.

If $f,g:X\to\mathbb{R}$ then, $f\wedge g=\frac{f+g-|f-g|}{2}$ and $f\vee g=\frac{f+g+|f-g|}{2}$

P.S. common notation $f\vee g=\min\{f,g\}$ and simlilarly for the other. What it means is "take the bigger value of $f$ or $g$ at x"
April 9th 2010, 03:36 PM
sfspitfire23

Quote:

Originally Posted by Drexel28

If $f,g:X\to\mathbb{R}$ then, $f\wedge g=\frac{f+g-|f-g|}{2}$ and $f\vee g=\frac{f+g+|f-g|}{2}$

P.S. common notation $f\vee g=\min\{f,g\}$ and simlilarly for the other. What it means is "take the bigger value of $f$ or $g$ at x"

Thank you Drexel. However, I'm having trouble seeing where you got $f\wedge g=\frac{f+g-|f-g|}{2}$ and $f\vee g=\frac{f+g+|f-g|}{2}$ . If you could explain to me how these came to be? (something with the definition of continuity? I haven't gotten to any value theorems yet)
April 9th 2010, 03:46 PM
Drexel28

Quote:

Originally Posted by sfspitfire23

Thank you Drexel. However, I'm having trouble seeing where you got $f\wedge g=\frac{f+g-|f-g|}{2}$ and $f\vee g=\frac{f+g+|f-g|}{2}$ . If you could explain to me how these came to be? (something with the definition of continuity? I haven't gotten to any value theorems yet)

No. That's the definition of min and max. Just try it out. "If $f$ is bigger than $f=\max\{f,g\}=\frac{f+g+|f-g|}{2}=\frac{f+g+f-g}{2}=\frac{2f}{2}=f$ ..."
April 9th 2010, 04:19 PM
sfspitfire23

I guess this is a hole in my Analysis. How would you derive f+g+|f-g|/2 from max{f,g}? Thank you for your patience...I'm trying to pick up on it
April 9th 2010, 04:23 PM
Drexel28

Quote:

Originally Posted by sfspitfire23

I guess this is a hole in my Analysis. How would you derive f+g+|f-g|/2 from max{f,g}? Thank you for your patience...I'm trying to pick up on it

I just told you.

$\max\{f,g\}=\begin{cases}f(x) &\mbox{if} \quad f(x)\geqslant g(x) \\ g(x) & \mbox{if} \quad g(x)\geqslant f(x)\end{cases}$ .

If $f(x)\geqslant g(x)$ then $f(x)-g(x)\geqslant 0$ and so [tex]|f(x)-g(x)|=f(x)-g(x)[/math\ so $\frac{f+g+|f-g|}{2}=\frac{f+g+f-g}{2}=\frac{2f}{2}=f$ . If $g(x)\geqslant f(x)$ then $|f-g|=|g-f|=g-f$ and so $\frac{f+g+|f-g|}{2}=\frac{f+g+g-f}{2}=\frac{2g}{2}=g$