Natural Log : seems as a discontinous function

hms.tech

The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c). In mathematical notation, this is written as
[itex]lim_{x\rightarrow c}[/itex] f(x) = f(c) from the positive and negative sides .

For ln(x) (the natural log of x), as x[itex]\rightarrow[/itex]0 , ln(x) approaches -[itex]\infty[/itex]

Hence I would stand by the notion that ln(x) is not continuous since at x=0, the function is not defined .

Also, the graph does not exist in x<0 domain ; so ln(x) can never approach -[itex]\infty[/itex] from the -ve "x" side .

pwsnafu

A function is continuous if it continuous at every point in its domain. The domain of natural logarithm (when treated as a real function) is ##(0,\infty)##. Zero is not part of the domain of log.

rasmhop

The part you seem to forget is "some point c of its domain". 0 is not in the domain of ln so this is why your observation is not a problem. The domain of ln is the positive numbers.

It is however an interesting observation in its own right and it implies that you cannot possibly define ln(0) in such a way that the extension would be continuous.

hms.tech

extending the idea you proposed, would it be logical to assume that each and every log(x) function has a domain (0,infinity) ?

rasmhop

Yes (slight lie, see below). Every log is defined precisely on the positive numbers.



This is a slight lie because in complex analysis we actually extend log and define it for many complex numbers. However we will never be able to include 0 in the domain (without making it discontinuous), and this little side remark is not important to you right now.