I’m studying measure theory, and my lecture notes state the following result:

Corollary 3.5. Let D be a Borel subset of ℝᵈ, and let f map D into the extended real line ℝ̄ = ℝ ∪ {−∞, +∞}. Then f is Borel measurable if and only if, for every c ∈ ℝ̄, the set {x ∈ D : f(x) < c} is a Borel set. The same criterion holds for Lebesgue measurability.

I picture c as a horizontal line across the graph of f. What trips me up is the phrase “for every c.” I mean, I can always choose c so low, such that for any function f, no x is below c, so basically only the empty set ever satisifes that criterion, which cannot be right, since then it would be worthless. So there is something I just don't understand. Please help me.