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Re: Limit[f[x], x->a] vs. f[a]. When are they equal?

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On 26 Apr 2011, at 10:43, Richard Fateman wrote:

> Hm. continuity  (excerpt from wikipedia, but whatever)
>
> The limit of f(x) as x approaches c through domain of f does exist and
> is equal to f(c); in mathematical notation, \lim_{x \to c}{f(x)} = f(c).
> If the point c in the domain of f is not a limit point of the domain,
> then this condition is vacuously true, since x cannot approach c through
> values not equal c. Thus, for example, every function whose domain is
> the set of all integers is continuous.
>
> ........
> There is now a kind of semantic gap, it seems to me. There are, perhaps,
> several gaps.


Here is another point worth (perhaps) commenting upon. The function 1/x defined on the real line excluding the point 0 is continuous everywhere because it is continuous wherever it is defined. Sometimes however mathematicians do say that this function "is not continuous at 0" and when they do so they mean that there is no way to extend its domain of definition so as to include 0 and make the new function continuous. This is unlike the function Sin[x]/x which is also continuous everywhere on the real line minus 0, but can be continuously extended to include 0 (and then it is nowadays called Sinc).

It is correct to say that Sin[1/z] is not continuous at zero because there is no way to give it a value there that would make it continuous. In the same sense Sin[z] is not continuous at ComplexInfinity: it cannot be continuously extended to the Riemann sphere. On the other hand the function 1/x canbe and hence:

1/ComplexInfinity

0

or, if you prefer

In[34]:= Limit[1/x, x -> ComplexInfinity, Direction -> 1]

Out[34]= 0

or

In[35]:= Limit[1/x, x -> ComplexInfinity, Direction -> I]

Out[35]= 0

etc.

The functions x and  1/x are "continuous at Infinity" as is every meromorphic function. They define continuous mappings from the Riemann sphere to its elf.  The function Sin[x] is not meromorphic and cannot be continuously extended to the Riemann sphere.

Andrzej Kozlowski

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Re: Limit[f[x], x->a] vs. f[a]. When are they equal? Andrzej Kozlowski <akoz@mimuw.edu.pl> - 2011-04-27 09:36 +0000

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