- #1

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x

^{2}= x

^{3}/ x but x

^{2}is defined for all x and equals zero at x=0 but what happens for x

^{3}/ x at x=o ? Is it defined at x=o ? Does it equal zero ? If not what is causing this anomaly ?

Thanks

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- Thread starter dyn
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- #1

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x

Thanks

- #2

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- #3

fresh_42

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##x \longmapsto x ## is everywhere continuous, ##x\longmapsto \dfrac{x^3}{x^2}## is not; at ##x=0\,.##

x^{2}= x^{3}/ x but x^{2}is defined for all x and equals zero at x=0 but what happens for x^{3}/ x at x=o ? Is it defined at x=o ? Does it equal zero ? If not what is causing this anomaly ?

Thanks

Although this is a removable singularity, it still is one, a gap. Algebraically division by zero isn't defined, simply because zero isn't part of any multiplicative group. The question never arises. It's like discussing the height of an apple tree on the moon.

- #4

FactChecker

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- #5

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I have seen the following statement in textbooks " x

- #6

fresh_42

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Strictly, yes. But as it isn't defined for ##x=0## it is implicitly clear. As long as you don't want to write unnecessary additional lines, just leave it. Who writes ##x\geq 0## if he uses ##\sqrt{x}\,?## This is simply self-evident, resp. clear by context. But logically, the domain of ##x## needs to be mentioned in general, such that we know what the function really is. But in your post it was pretty clear what you meant even without it.So x = x^{3}/ x^{2}is not a correct statement on its own ? It needs the addition of the statement " x not equal to 0 " ?

See above. It is simply not necessary as long as you don't write a book on logic. It's like mocking about a Pizza guy not telling you it's hot. However, if you talk about specific functions, you better say where and how they are defined. E.g. you could defineI have seen the following statement in textbooks " x^{n}/ x^{m}= x^{n-m}" with no mention of " x not equal to 0 ". Are they just being lazy and missing out the " x not equal to 0 " statement ?

$$f(x) = \begin{cases}\dfrac{x^3}{x^2} \,&,\, x\neq 0 \\ 0\,&\,,x=0\end{cases}$$

or simply

$$

f(x)= \dfrac{x^3}{x^2}\; , \;x\neq 0

$$

which will be two different functions. So as always with written things, it depends on what you want to express. In post #1 and the example you gave it isn't necessary. Mocking about it is nit-picking.

- #7

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I want to point out that I wasn't mocking about it ; I just wanted clarity. A lot of time some parts of maths seem like nit-picking to me but as plenty of people on here point out the finer points can be important.

- #8

FactChecker

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I think that should be stated more carefully. Certainly, if ##x = -y^2+y+5## one would not use ##\sqrt {x}## without specifying that ##-y^2+y+5 = x \ge 0 ## and determining what the corresponding valid values of ##y## are.Who writes ##x\geq 0## if he uses ##\sqrt x##? This is simply self-evident, resp. clear by context.

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