Question: The first equality here cannot be justified without an appeal to the continuity of P, Theorem 1.54. For the second part of the theorem, if

The first equality here cannot be justified without an appeal to the continuity of P, Theorem 1.54. For the second part of the theorem, if a ≤

b, then P(a ≤ X ≤

b) = P(a < X ≤

b) by (5.28)

= FX

(b) − FX

(a) by (5.10)

=

Z b a

fX (u) du. 2

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