Question regarding the continuity of functions

[new_at_math]

Homework Statement

so a function was only continuous if and only if lim x ---> a = f(a)

but I read some other books saying a functions continuity can only be judged based on their allowed domains. Does that mean the first definition fails if the domains of where the rule fails are explicitly stated or when using a "familiar" parent graph?

Homework Equations

refer to above

The Attempt at a Solution

the book I read,the humongus of calculus, said but I read some other books saying a functions continuity can only be judged based on their allowed domains. The book doesn`t seem to serious with math. I don`t know how to approach this. I think the book meant to say that this only applies to continuity over an interval rather than the meaning a whole function was continuous.

 

[Dick]

Homework Statement

so a function was only continuous if and only if lim x ---> a = f(a)

but I read some other books saying a functions continuity can only be judged based on their allowed domains. Does that mean the first definition fails if the domains of where the rule fails are explicitly stated or when using a "familiar" parent graph?

Homework Equations

refer to above

The Attempt at a Solution

the book I read,the humongus of calculus, said but I read some other books saying a functions continuity can only be judged based on their allowed domains. The book doesn`t seem to serious with math. I don`t know how to approach this. I think the book meant to say that this only applies to continuity over an interval rather than the meaning a whole function was continuous.

Your first statement defines continuity at x=a. f needs to be defined in an open interval around x=a for that to work. Other than that I don't know what the 'allowed domains' thing is about. Can you quote something? Being continuous at a point is certainly different from being continuous everywhere. Is that what you are getting at?

 

[Mark44]

Homework Statement

so a function was only continuous if and only if lim x ---> a = f(a)

As written, this doesn't mean what you're trying to say - you left part out. It should be like so:
$$\lim_{x \to a}f(x) = f(a)$$

but I read some other books saying a functions continuity can only be judged based on their allowed domains. Does that mean the first definition fails if the domains of where the rule fails are explicitly stated or when using a "familiar" parent graph?

Homework Equations

refer to above

The Attempt at a Solution

the book I read,the humongus of calculus, said but I read some other books saying a functions continuity can only be judged based on their allowed domains. The book doesn`t seem to serious with math. I don`t know how to approach this. I think the book meant to say that this only applies to continuity over an interval rather than the meaning a whole function was continuous.

 

[eumyang]

so a function was only continuous if and only if lim x ---> a = f(a)

If this is supposed to define continuity of a function at a, you're missing some details. If I remember correctly, for a function to be continuous at a, three conditions have to be satisfied:
1. $f(a)$ is defined.
2. $\lim_{x \to a}f(x)$ exists.
3. $\lim_{x \to a}f(x) = f(a)$

For a function to be continuous in some open interval (a, b), the function has to be continuous at every point in the interval.

 

[pasmith]

Homework Statement

so a function was only continuous if and only if lim x ---> a = f(a)

$$\lim_{x \to a} f(x) = f(a)$$

but I read some other books saying a functions continuity can only be judged based on their allowed domains. Does that mean the first definition fails if the domains of where the rule fails are explicitly stated or when using a "familiar" parent graph?

The domain of a function is part of its definition. A function is defined at all points of its domain and is undefined everywhere else. A function can't be continuous at a point outside its domain because its value is not defined there.

The formal definition of continuity of a real function is that $f: U \subset \mathbb{R} \to \mathbb{R}$ is continuous at $a \in U$ if and only if for all $\epsilon > 0$ there exists $\delta > 0$ such that for all $x \in U$, if $|x - a| < \delta$ then $|f(x) - f(a)| < \epsilon$.

The definition does not require that $(a - \delta, a + \delta) \subset U$. Indeed it may happen that $U \cap (a - \delta, a + \delta) = \{a\}$, in which case $f$ being continuous as $a$ tells you nothing about its values at any other points in $U$.

Thus, for example, any $f: U = (-\infty, 1] \cup \{2\} \cup [3, \infty) \to \mathbb{R}$ is continuous at 2, because for all $x \in U$, if $|x - 2| < \frac12$ then $x = 2$. Thus for all $x \in U$, if $|x - 2| < \frac12$ then $|f(x) - f(2)| = 0$.