What is meant by f being continous in possible isolated points in its domain?

Asked May 25 '24 at 22:36

Active May 25 '24 at 23:03

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The definition of a continuous function in my textbook is given as follows:

$f$ being continuous in a accumulation point $a\in D_f$ means that:

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

We count $f$ as continuous in possible isolated points $a\in D_f$

I do understand the first statement about $a$ being a accumulation point. But I dont understand the last statement:

We count $f$ as continuous in possible isolated points $a\in D_f$

Could someone explain what is meant by this with an example?

edited May 25 '24 at 23:03

asked May 25 '24 at 22:36

a_floating_point

1

I know what is intended by the 2nd statement, but it's worded very poorly. The intended meaning is: "If $a$ is an isolated point of $D_f,$ then $f$ is continuous at $a.$" In other words, by definition $f$ is (automatically) continuous at every isolated point of its domain (regardless of whatever formula might be used to define $f$ and regardless of the values of $f$ at other points of its domain). – Dave L. Renfro May 25 '24 at 23:11
Does this answer your question? Can a function with just one point in its domain be continuous? – Sam May 25 '24 at 23:38

What is meant by f being continous in possible isolated points in its domain?

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