Defintion of a limit of a function by Serge Lang - unusual??!!

I am working through Serge Lang's book "Undergraduate Analysis" (Second Edition)

On pagfe 42 he defines the limit of a function as follows:

"We shall say that the limit of f(x) as x approaches a exists if there exists a number L having the following property. Given $\displaystyle \epsilon$, there exists a number $\displaystyle \delta$ > 0 such that for all x $\displaystyle \in$ S satisfying

|x - a| < $\displaystyle \delta$

we have

|f(x) - L| < $\displaystyle \epsilon$ {see below for some

obvious and immediate consequences}

This definition seems at odds with the definition in Apostal and other texts snice they do not allow x to actually assume the value a and write something equivalent to

0 < |x - a| < $\displaystyle \delta$

Am I right in assuming Lang's definition differs?

If so - are there significant consequences for theorems - I mean does one constantly have to be careful over this matter?

{Note further that on page 43 Lang writes:

"Next, suppose a is an element of S. We consider any function f on S.

Then the limit $\displaystyle lim x_{\rightarrow a} f(x) $ exists.

We contend that it must be equal to f(a)"

Surely (as a consequence of Lang's defn) this is not the usual conclusion!

Yet further ... on page 44 Lang writes:

"Define g on S by g(x) = x if x ne 0 and g(0) = 1. Then $\displaystyle lim_{x \rightarrow 0} f(x) $ does not exist. Again with the defn of Apostal and others $\displaystyle lim_{x \rightarrow 0} f(x) $ would be equal to 0 even though the value of the function at 0 is 1."

Is my reasoning correct?

Is it right to be alarmed at the possible consequences of this different definition for being able to follow mainstream analysis? Should I switch to another text? }

Bernhard

Defintion of a limit of a function by Serge Lang - unusual??!!

Thanks CB

I presume that, under Serge Lang's definition, if the Set S includes 0 and $\displaystyle \frac{sine\ x}{x}$ is defined as 1 at 0 then $\displaystyle lim_{x \rightarrow 0}\frac{sine\ x}{x}$ = 1. If $\displaystyle \frac{sine\ x}{x}$ is defined as something else at 0 then according to Lang the limit does not exist.

On the other hand if the set S does not include 0 [but includes all points around 0] then the limit exists and is equal to 1.

Does that sound right?