If P(X < a) = b then is P(f(X) < f(a)) = b?

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If f(x) is a monotonic increasing function, then does P(X < a) = b imply P(f(x) < f(a)) = b ? My intuition says it's true but I cannot prove the case nor find the name of the theorem.
mathematical-statistics function measure-theory
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asked 1 hour ago
Linsu Han
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By "monotonic increasing" do you mean strictly increasing (if $a ...