If $X$ is a continuous random variable with support $A$, does this imply that the cdf of $X$ is strictly increasing on $A$?
My guess is yes. But just in case, let me know if you can think of any counterexamples.
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Sign up to join this communityA counter example is made by taking the cdf as the Cantor distribution $\mathfrak c(\cdot)$ on $[0,1]$, whose support is the Cantor set $\mathfrak C$:
Assuming the density exists, and ignoring the boundary points, a function strictly increases if its derivative is strictly positive. CDF’s derivative is PDF and it is strictly positive in the support. So, CDF must strictly increase in the support.