One-sided continuity.

Definition 5.4.1   The function is left continuous (resp. right) at $ x_0$ if it has a left limit (resp. right limit) at $ x_0$ and if this limit is equal to $ f(x_0)$ .

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 $ f$ be a function defined over a neighborhood of the real number $ x_0$ . The function $ f$ is continuous at $ x_0$ if, and only if, the two conditions hold:
  1. the function $ f$ is left-continuous at $ x_0$ ;
  2. the function $ f$ is right-continuous at $ x_0$ .

Example 5.4.3   Take $ f(x)=\vert x\vert$ . We have:
(i)
If $ x \leq 0$ , then $ f(x)=-x$ . It follows that $ \underset{x \underset{<}{\rightarrow} 0}{\lim} f(x)=0=f(0)$ ; thus $ f$ is left continuous at 0.
(ii)
If $ x \geq 0$ , then $ f(x)=x$ . It follows that $ \underset{x \underset{>}{\rightarrow} 0}{\lim} f(x)=0=f(0)$ ; thus $ f$ is right continuous at 0.
Thus, $ f$ is continuous at 0; see Figure 4.

Figure 4: The absolute value function.
\begin{figure}\centering
\mbox{\epsfig{file=AbsoluteValue.eps,height=4.5cm}}\end{figure}

Example 5.4.4   Let $ f$ be given as follows:

\begin{displaymath}\begin{cases}f(x)=2x^2-3x+5, x \leq 0  f(x)=\frac {2x+5}{x+1} \end{cases}.\end{displaymath}    

Let us compute the one-sided limits of $ f$ at 0:

$\displaystyle \underset{x \underset{<}{\rightarrow} 0}{\lim} f(x)$ $\displaystyle = \underset{x \underset{<}{\rightarrow} 0}{\lim} (2x^2-3x+5)=5$    
$\displaystyle \underset{x \underset{>}{\rightarrow} 0}{\lim} f(x)$ $\displaystyle = \underset{x \underset{>}{\rightarrow} 0}{\lim} \frac {2x+5}{x+1}=5.$    

As $ f(0)=5$ , this shows that $ f$ 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]$ . Let $ n \in \mathbb{Z}$ . Then the following holds:

$\displaystyle \underset{x \underset{<}{\rightarrow} n}{\lim} f(x)= n-1$   and$\displaystyle \qquad \underset{x \underset{>}{\rightarrow} n}{\lim} f(x) = n.$    

As we know that $ f(n)=n$ , we conclude that $ f$ is right-continuous at $ n$ but not left-continuous at $ n$ .

Noah Dana-Picard 2007-12-28