What's the midpoint of $[a,b)$? While the midpoint of either $[a,b]$ or $(a,b)$ is $\dfrac{a+b}{2}$, what's the midpoint of $[a,b)$ if it has been defined at all?
 A: It depends on how you want to define the so called "mid point" of an interval. One can simply define the mid point of an interval to be $\frac{a+b}2$ for any interval of the form $(a,b), (a,b], [a,b)$ or $[a,b]$. This definition would serve most general purposes quite well since a single point has measure zero. 
However, suppose you want the define the mid point to have the property such that 

for an interval $I$, $m$ is called the midpoint of $I$ if for any $x\in I$ there exist $x'\in I$ such that $x-m=m-x'$

then it's provable that $[a,b)$ has no mid point. (Hint: if such $m$ exists then $I$ must be of the form $(m-d,m+d)$ or $[m-d,m+d]$)
A: It's still $\displaystyle\frac{a+b}{2}$.
A: It's the same, $\dfrac{a+b}{2}$.  It's still the point equidistance from the end points.  (The "end points" don't actually have to be in the interval itself.)
A: Here's one possible definition. Let $A$ denote a measurable subset of $\mathbb{R}$. Then a midpoint of $A$ is an $x \in \mathbb{R}$ such that $\mu(A \cap (\infty,x]) = \mu(A \cap [x,\infty)),$ where $\mu$ denote the Lebesgue measure. Observe that a measurable subset of $\mathbb{R}$ may have many midpoints (for example, $[0,1] \cup [2,3]$) or none (for example, $[0,\infty).$) But all three of $[a,b]$, $[a,b)$ and $(a,b)$ have precisely one midpoint, namely $(a+b)/2$.
A: I would say:
Midpoint=$(a+b)/2$. Think this way, if b) is infinitesimally close to b] the midpoint is the same as (a,b) or [a,b]. For example, if b=2.0 and you get to 1.99999999999....... the midpoint is the same as [a,b].
A: The standard definition of the midpoint is, "The point on a line segment dividing it into two segments of equal length" or the equivalent. As a result, the midpoint is found by the formula $\frac{a + b}{2}$ for all types of intervals. 


*

*https://en.wikipedia.org/wiki/Midpoint

*http://mathworld.wolfram.com/Midpoint.html
