[hist-analytic] Clarification: Re: Elizabeth Anscombe's Intention (New "Look Inside" feature)

Roger Bishop Jones rbj at rbjones.com
Thu Jun 17 00:52:37 EDT 2010


On Thursday 17 Jun 2010 03:21, Baynesr at comcast.net wrote:
> There is a simpler way I should have replied to your
>  statement
> 
> "Lim f(x) = fa
> x -> a
> 
> This is a definition of continuity"
> 
> What you say looks right, but here is where, I think,
>  it's wrong. It is true that in the case of 'Lim f(x) =
>  f(a) '
> x->a
> 
> the function IS continuous, BUT only at that point, a.

Yes.
To be more explicit I should have said something like:
"this is the condition for continuity of f at a"

This is what wikipedia says

In general, we say that the function f is continuous at some 
point c of its domain if, and only if, the following holds:
The limit of f(x) as x approaches c through domain of f does 
exist and is equal to f(c); in mathematical notation, ...
>  But for a function to be continuous it must be
>  continuous at all points. So the formula entails
>  continuity but only at a point; it does not therefore
>  define 'continuity'. Continuity of the function requires
>  of the function continuity at all points.The formula is
>  sufficient but not necessary for continuity, so it's not
>  a definition of continuity. I think that is the best way
>  to state where we might disagree.

I don't disagree!

Roger
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