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When introducing the concept of continuous random variables, my textbook states the following:

If $X$ is a continuous random variable, then $P(a \le X \le b) = P(a < X \le b) = P(a \le X < b) = P(a < X < b)$.

However, it does not give a justification for why this is true.

I would greatly appreciate it if people could please take the time to explain why this is true.

The Pointer
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  • Because $P(X=c)=0$ for every constant $c\in\mathbb R$ (as continuous random variable $X$ has a continuous CDF). – drhab May 27 '17 at 10:18
  • For a continuous RV the inequality or equality of ranges do not matter precisely because probability at a point vanishes in this case. – StubbornAtom May 27 '17 at 10:19
  • @jvdhooft Thanks for the response. I understand why the probability density function $= 0$ for any specific value: It has to do with the property of continuity. However, I don't understand how this answers my specific question? – The Pointer May 27 '17 at 10:20
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    @ThePointer $P(a \le X \le b) = P(a \lt X \le b) + P(a = X)$. See where I'm going with this? – jvdhooft May 27 '17 at 10:23
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    Do you realize that e.g. $P(a\leq X\leq b)=P(X=a)+P(a<X\leq b)$ so that $P(X=a)=0$ explains it? Or have you a difficulty with $P(X=a)=0$ then? Do not involve a PDF because it might not exist. – drhab May 27 '17 at 10:23
  • Ahh, this makes complete sense. Thank you all! – The Pointer May 27 '17 at 10:42

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