A random variable X is continuous if and only if it's CDF F is continuous.But how can we prove F to be uniform continuous?
2026-03-25 04:39:26.1774413566
Uniform continuity of the CDF of a continuous Random variable.
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If $F:\Bbb R\to\Bbb R$ is continuous and has (finite) limits at $\pm\infty$ then $F$ is uniformly continuous.
You should concoct your own proof of this, starting with "let $\epsilon>0$". A sort of high-level proof, purposely chosen so as not to give you that proof you should be looking for: $F$ extends to a continuous function $F:[-\infty,\infty]\to\Bbb R$, and $[-\infty,\infty]$ is a compact metric space...