But what is a continuous function?

Question

I have a very basic problem. I am confused about "continuous function" term.

What really is a continuous function? A function that is continuous for all of its domain or for all real numbers?

Let's say:

$\ln|x|$ - the graph clearly says it's continuous for all real numbers except for $0$ which is not part of the domain. So is this function continuous or not? I could say same about $\tan{x}$ or $\frac{x+1}{x}$

And also what about:

$\ln{x}$ - the graph clearly says it's continuous for all of its domain: $(0; \infty)$ - so is this $f$ continuous or not?

Thanks for clarification.

Answer 1

Mathematicians (but not all calculus books) mean "continuous at every point of its domain" when they say a function is "continuous." The functions $f(x) = 1/x$ and $f(x)=\ln x$ are continuous functions.

Answer 2

"Continuous" is not, in and of itself, a property of a function. You have to talk about being continuous at a given point, or on a collection of points as you have above.

It is generally safe to assume that if somebody leaves off the set, they intend to say that the function is continuous on its domain (as both of your examples are); but, I tend to believe that explicit is better than implicit.

Answer 3

The exact answer depends on your chosen definition of "function" (there is more than one). For most uses, a function is regarded as being continuous on an interval $(a,b)$ if for every number $c$ in $(a,b)$, $f(x)=\lim_{x\to c} f(x)$.

In your example $f(x)=\ln{x}$ is continuous on the interval $(0,\infty)$ and either undefined or complex/multivalued everywhere else, depending on whether you consider the codomain (range) of $f$ to include the complex numbers or not.

In other words, no function is ever just 'continuous' - it is continuous within an interval (which may or may not be its domain).

Answer 4

It is, but what you are looking for might be the notion of continous extension.

Here both $\frac{1}{⋅}$ and $\log$ are continuous in the sense that they are (pointwise) continuous on their respective domains, as is $$ \left| \begin{array}{lll} f : &\mathbb{R}^* ⟶ \mathbb{R}\\ & x \longmapsto x² \end{array} \right. $$ The difference is that there exists $g: \mathbb{R}→\mathbb{R}$ continuous such that $g(x)=f(x)$ for all $x∈\mathbb{R}^*$ — so $f$ has a continous extension on $\mathbb{R}$ — whereas $\frac{1}{⋅}$ has no such thing.