3

Very often, we define the limit of a function as $0 < |x -a|< \delta \implies |f(x) - L|< \epsilon$.

A lot of times we don't let $x$ equal $a$, and the reason for this is clear in the case of discontinuity. However, the book I am reading also doesn't allow $x = a$ for continuous functions.

John Bentin
  • 18,454
Lemon
  • 12,664
  • One way to define a continuous function is to say that $\lim_{x\to a}f(x)=f(a)$ where $f(a)$ is a real number. If your book defines the a function to be continuous at $a$ when the limit exists, equals $L$ and $f(a)=L$ then this is the same. – Kris Williams Jul 03 '13 at 19:08
  • “Not allowing $x\neq a$ ” means that $x$ is fixed to be $a$. It is clear that this is not the meaning intended by the OP. – John Bentin Jul 10 '23 at 15:46

3 Answers3

4

The function may be undefined at $x=a$. For example, take $a=0$ and let $f(x)=(\sin x)/x\;(x\in \Bbb R;x\neq 0).$ Then the limit is well defined at $0$, even though $f$ is not. (Of course, in this case, we can easily and naturally extend the domain of $f$ by using the limit.)

John Bentin
  • 18,454
4

While I agree that you could let $x=a$ in some cases, forbidding this reinforces the idea that a limit is a property that holds near a point, and is defined in these terms: i.e. a limit $L$ exists if it is the unique number (real or complex depending upon the context) that is arbitrarily close to $f(x)$ whenever $x$ is arbitrarily close to $a$ but not equal. We define the number $L$ to be the limit if it has precisely this property. Deleting $a$ does not change the limit if it exists.

  • But what happens if we have a smooth function? The book I am reading suggests we can't even have $x \neq 0$ in any case – Lemon Jul 03 '13 at 23:40
  • The point is that even if the function is defined to be the limit at $x=a$, which would be true for a smooth function, this is irrelevant for what the limit is. The limit exists only if it is close to all values in the range if we are close to $a$ for all values in the domain except for a. In fact, take the opposite extreme: define $f$ at $a$ to NOT equal the limit, which is perfectly fine. But the limit would be unaffected by this strange definition. This gives even more impetus to looking at a deleted neighborhood. –  Jul 04 '13 at 00:03
0

Note that LIMIT is a DYNAMIC PROCESS, not STATIC equilibrium. Therefore the value of f precisely at the point a is irrelevant. It is important to understand that the limit of a function is the condition on the behavior of a function in a deleted neighborhood of a point, rather than a condition on its value at a particular point.

Will be revisited after I find my old book note when I made at the first sight of limit many years ago.

MathArt
  • 1,053