Left/Right Continuity

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The limit of a continuous function at an endpoint to determine continuity at endpoints.

At this point we have a small problem. We’ve discussed continuity for points in open intervals; in particular, for values where you can look at points “nearby” the point of interest on both the right and left sides. Consider functions such as $\sqrt {x}$ , the natural domain is $0\leq x <\infty$ . This is not an open interval. What does it mean to say that $\sqrt {x}$ is continuous at $0$ when $\sqrt {x}$ is not defined for $x<0$ ? To get us out of this quagmire, we need a new definition:

A function $f$ is left continuous at a point $a$ if $\lim _{x\rightarrow a^-} f(x) = f(a)$ .

A function $f$ is right continuous at a point $a$ if $\lim _{x\rightarrow a^+} f(x) = f(a)$ .

This allows us to talk about continuity on closed and half-closed intervals.

A function $f$ is

continuous on a closed interval $[a,b]$ if $f$ is continuous on $(a,b)$ , right continuous at $a$ , and left continuous at $b$ ;
continuous on a half-closed interval $[a,b)$ if $f$ is continuous on $(a,b)$ and right continuous at $a$ ;
continuous on a half-closed interval $(a,b]$ if $f$ is $f$ is continuous on $(a,b)$ and left continuous at $b$ .

Intuitively this means that a function is called continuous if it is continuous at all the points in its domain; with the understanding that we mean left or right continuous for “end-points” of the domain (if the end-point is included in the domain).

1 : Here we give the graph of a function defined on $[0,10]$ .

Select all intervals for which the following statement is true.

The function $f$ is continuous on the interval $I$ .

$I=[0,10]$ $I=[0,4]$ $I=[4,6]$ $I=[4,6)$ $I= (4,6]$ $I= (4,6)$ $I=(6,10]$ $I=[6,10)$

Notice that our function is left continuous at $x=4$ so we can include $4$ in the interval $[0,4]$ . Four is not included in the interval $(4,6)$ because our function is not right continuous at $x=4$ . Similarly, our function is neither right or left continuous at $x=6$ , so $6$ is not included in any intervals. Our function is left continuous at $x=0$ and right continuous at $x=10$ so we included these endpoints in our intervals.

Notice that if we didn’t have the definitions for left and right continuity, then we would always have to include the endpoints of a domain as points of discontinuity; which should seem intuitively wrong. Thus these definitions are really a way to fill in the holes of our definition that our intuition have detected. This is a common process in mathematics; to build definitions in a way to fully encompass what we are trying to describe; and sometimes that requires small sub-definitions - really an extension of the core idea - to fill some special cases.