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## Main Question or Discussion Point

Let the random variable X have the probability density function f(x). Suppose f(x) is

continuous over its domain and Var[X] is bounded away from zero: 0 < c < Var[X].

Claim: f(x) is bounded over its domain.

Is this claim true?

I don't think a counterexample like X ~ ChiSq_1 applies because, while f(0) is not bounded, the domain is x > 0. The question of continuity at 0 doesn't arise since 0 is not part of the domain, and therefore the lack of boundedness at 0 isn't relevant. But for any given c > 0, I can find an M such that f(c) < M. This example does show, however, why one can't claim "uniformly bounded."

Similarly, a degenerate r.v. (all probability mass at the point X = c, say) is continuous (because it is discrete), but the variance is not bounded away from zero.

So the question remains: is the claim true? How might one prove it?

Thanks.

continuous over its domain and Var[X] is bounded away from zero: 0 < c < Var[X].

Claim: f(x) is bounded over its domain.

Is this claim true?

I don't think a counterexample like X ~ ChiSq_1 applies because, while f(0) is not bounded, the domain is x > 0. The question of continuity at 0 doesn't arise since 0 is not part of the domain, and therefore the lack of boundedness at 0 isn't relevant. But for any given c > 0, I can find an M such that f(c) < M. This example does show, however, why one can't claim "uniformly bounded."

Similarly, a degenerate r.v. (all probability mass at the point X = c, say) is continuous (because it is discrete), but the variance is not bounded away from zero.

So the question remains: is the claim true? How might one prove it?

Thanks.