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 December 7th, 2010, 12:47 PM #1 Newbie   Joined: Sep 2010 Posts: 18 Thanks: 0 math analysis show that f(x) is continuous on R but not differentiable on R f(x) = (x^2 -1)sin(1/x-1) if x don't equal 1 f(x) = o , if x=1 2/ let f:A?B and C ? B , D ? B prove that ƒ -¹ (C/D) = ƒ -¹ (C)/ƒ -¹(D)
December 9th, 2010, 06:33 PM   #2
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Re: math analysis

Quote:
 Originally Posted by alice 9 show that f(x) is continuous on R but not differentiable on R f(x) = (x^2 -1)sin(1/x-1) if x don't equal 1 f(x) = o , if x=1 2/ let f:A?B and C ? B , D ? B prove that ƒ -¹ (C/D) = ƒ -¹ (C)/ƒ -¹(D)
How can we know that exercise No 2 is correct ??

IS $f(x)= (x^2-1)sin(\frac{1}{x}-1)$ in exercise No 1 ??

 December 10th, 2010, 11:58 AM #3 Newbie   Joined: Sep 2010 Posts: 18 Thanks: 0 Re: math analysis fore q 1 its (x^2 - 1) sin 1 over (x-1)
December 11th, 2010, 06:07 PM   #4
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Re: math analysis

Quote:
 Originally Posted by alice 9 fore q 1 its (x^2 - 1) sin 1 over (x-1)
To prove that $f^{-1}(C/D)= f^{-1}(C)/f^{-1}(D)$ we must prove that left hand side is a subset of the right hand side and conversely.

But for the left hand side to be a subset of the right hand side we must have that:

if $x\in f^{-1}(C/D)$,THEN $x\in (f^{-1}(C)/f^{-1}(D)$).

.........................................OR....... ......(By using the definition of preimage).....................

$x\in A$ and ($f(x)\in C$ and $f(x)\notin D$) $\Longrightarrow x\in f(x)^{-1}(C)$ and $x\notin f(x)^{-1}(D)$.

.................................................. ...OR............................................. .........................

$x\in A$ and ($f(x)\in C$ and $f(x)\notin D$) $\Longrightarrow$$[ (x\in A$and$f(x)\in C)$ and $(x\in A\Longrightarrow f(x)\notin D)]$

Now can you curry on from here??

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