the defining properties of a cdf
Not every curve can be a cumulative distribution function. There is a short checklist that captures exactly what it means to accumulate probability honestly, and any function passing it is the cdf of some random variable, while any cdf must pass it. Knowing the list lets you spot a bogus F at a glance and build valid ones on purpose.
A cdf F must satisfy four conditions. First, it is non-decreasing: as you move right and let in more values, the accumulated probability can only stay the same or grow, never drop, because you are never subtracting probability. Second, its limits are 0 on the far left and 1 on the far right: as x goes to minus infinity, F(x) goes to 0 (no probability yet), and as x goes to plus infinity, F(x) goes to 1 (all of it). Third, it is right-continuous: at any value, F(x) equals the limit as you approach x from the right, F(x) = limit of F(x + h) as h decreases to 0. This right-continuity is forced by the <= in P(X <= x): the threshold value itself is included.
These properties have a real payoff. Right-continuity means that at a jump, the function takes the upper value, and the jump height equals P(X = x), the mass exactly at that point; on an interval where F is flat, no probability lives there. The far-left and far-right limits guarantee total probability 1. Honest subtlety: a cdf can jump (discrete part) and can have flat stretches, but it can never jump downward and never overshoot 1 or dip below 0. If a proposed F ever decreases, it is simply not a cdf.
F(x) = 0 for x < 0, F(x) = x for 0 <= x < 1, F(x) = 1 for x >= 1 is a valid cdf (the uniform on [0,1]): non-decreasing, limits 0 and 1, continuous. But G(x) = x^2 - x on [0,1] is not, because it dips below 0 and even decreases.
Non-decreasing, right-continuous, with limits 0 and 1: pass all four and you have a genuine cdf.
Right-continuity is the property students forget. It is not optional decoration; it is exactly what the <= in P(X <= x) demands, and it determines whether the value at a jump belongs to the step below or above.