This is a self-study question. The name of the book is called: Applied Statistics and Probability for Engineers by Montgomery and Runger. This problem is on page 73. It's exercise 3-41.
The entire problem is listed as the following:
Given the following cumulative distribution function:
$$
F(x)=\left\{\begin{matrix}
0 & & x<-10 \\
0.25 & & -10\leq x< 30 \\
0.75 & &30\leq x< 50 \\
1 & & 50 \leq x
\end{matrix}\right.
$$
Determine each of the probabilities:
a) $P(X<50)$ b) $P(0 \leq X < 10)$ c) $P(-10 < X <10) $… etc etc. The question I have is this:
Why does the following probability mass function evaluate to 0?
$$
P(0 \leq X < 10) = 0
$$
Isn't this set of outcomes a subset of $-10\leq x< 30$ and therefore should be evaluated to 0.25?
Best Answer
The cumulative probability distribution function $F_X(x)$ tells us how much probability mass there is to the left of $x$ or at $x$ for each $x$ on the real line. (The choice of notation, though almost universally used is truly dreadful for use in a classroom setting! How on earth does one read out aloud $F_X(x)$ or $P\{X\leq x\}$? F-sub-big X of little x? probability that random variable $X$ is no larger than lower-case x?) Formally, the value of $F_X(x)$ is just $P\{X \leq x\}$. As Glen_b's comment says, you really should start by sketching the function $F_X(x)$ at the very least.
When $X$ is a discrete random variable taking on values $x_1, x_2, \ldots$ with probabilities $p_1, p_2, \ldots $ respectively, a little thought (instead of rote memorization of the definition) reveals that $F_X(x)$ must be what can be described as a staircase function, increasing from $0$ to $1$ as $x$ increases, with steps of heights $p_1, p_2, \ldots$ at points $x_1, x_2, \ldots$ etc. The function is discontinuous at each $x_i$, and is constant in each interval $[x_i, x_{i+1})$ (please be sure to note the $[$ and $)$ in the description of the intervals). Note that $F_X(x_i)$ includes $p_i$ so that the value of $F_X(x)$ at the point $x=x_i$ (where the function is discontinuous) is the value on the right. Since you are studying from a text intended for engineers, you might find this written as $F_X(x) = F_X(x^+)$. Thus, $$F_X(x) = P\{X \leq x\} = F_X(x^+) ~ \text{and} ~ P\{X < x\} = F_X(x^-).$$
In fact, for any random variable (not necessarily a discrete random variable or an integer-valued random variable as in Rusan's answer) and for any real numbers $a$ and $b$ such that $a \leq b$, $$\begin{align} P\{a < X \leq b\} &= F_X(b^+) - F_X(a^+) = F_X(b)-F_X(a),\tag{1}\\ P\{a \leq X \leq b\} &= F_X(b^+) - F_X(a^-) = F_X(b) - F_X(a^-),\tag{2}\\ P\{a \leq X < b\} &= F_X(b^-) - F_X(a^-) = F_X(b^-) - F_X(a^-),\tag{3}\\ P\{a < X < b\} &= F_X(b^-) - F_X(a^+) = F_X(b^-)-F_X(a).\tag{4} \end{align}$$
For the special case when $b = a$, $(2)$ above becomes $$P\{X=a\} = F_X(a^+)-F_X(a^-),$$ that is, $P\{X=a\}$ is the jump (if any) in the value of $F_X(x)$ at $x=a$. If $F_X(x)$ is continuous at $x=a$, then $P\{X=a\}=0$.
With this as prologue, note that your given $F_X(x)$ is a staircase function with jumps of $\frac 14, \frac 12, \frac 14$ at $x=-10, 30, 50$ respectively; that is, $X$ takes on values $-10, 30, 50$ with probabilities $\frac 14, \frac 12, \frac 14$ respectively, and once you have that, the answers to the questions asked are easy to compute directly, or, if you prefer to read the $F_X(0^-)$ etc off the graph that you have drawn as you apply $(1)$-$(4)$, that is fine too.