Definition 5.4.1
The function is left continuous (resp. right) at

if it has a left limit (resp. right limit) at

and if this limit is equal to

.
For example, the integer part function is right continuous but not left continuous at 1 (v.s. figure
fig integer part 2).
Proposition 5.4.2
Let
be a function defined over a neighborhood of the real number
. The function
is continuous at
if, and only if, the two conditions hold:
- the function
is left-continuous at
;
- the function
is right-continuous at
.
Example 5.4.4
Let

be given as follows:
Let us compute the one-sided limits of

at 0:
As

, this shows that

is left-continuous at 0 and right-continuous at 0, thus is continuous at 0.
Example 5.4.5
Consider the "integer part" function, i.e.
![$ f(x)=[x]$](img558.png)
.
Let

. Then the following holds:
and |
|
As we know that

, we conclude that

is right-continuous at

but not left-continuous at

.
Noah Dana-Picard
2007-12-28