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A Cumulative Distribution Function (CDF) must be right-continuous by definition. Right-continuity means that the limit of the CDF as the variable approaches any point from the right equals the value of the CDF at that point. Formally, for a CDF function \(F_X(x)\), right-continuity is expressed as \lim_{{x \to c^+}} F_X(x) = F_X(c)\ for any point \c\. This condition ensures that there are no sudden jumps in the function when looking from the right side of the interval. In our exercise, the CDF \(F_X(x)\) satisfies this property across all intervals. Since \F_X(x)\ is defined piecewise, ensuring that the function aligns at the boundary values confirms right-continuity.

Checking the limits at infinity for a CDF is crucial as it tells us the behavior of the function at extreme values of \x\. For any CDF, the limit as \x\ approaches negative infinity should be 0, and the limit as \x\ approaches positive infinity should be 1. Mathematically, these properties are written as:
1. \lim_{{x \to -\tilde{\float{mathplus}}}} F_X(x) = 0\
2. \lim_{{x \to \tilde{\float{mathplus}}}} F_X(x) = 1\
This is because a CDF represents the probability that a random variable is less than or equal to \(x\). As \(x\) approaches negative infinity, this probability should indeed be zero, because no values can be below negative infinity. Similarly, as \(x\) approaches positive infinity, the probability reaches 1, embracing all possible outcomes.

Monotonicity in the context of CDFs means that the function is non-decreasing. In simpler terms, the CDF should either stay the same or increase as \(x\) increases鈥攏ever decrease. This property ensures that as we move along the x-axis from left to right, the probability captured by the CDF accumulates and doesn't reduce. Mathematically, for any two points \(a\) and \(b\) such that \(a < b\), it must hold that \(F_X(a) \leq F_X(b)\). In our exercise, the CDF is defined piecewise and we need to ensure that within each interval and at the boundaries, the function does not drop, satisfying this property of monotonicity.

The Probability Density Function (PDF) of a continuous random variable is the derivative of its CDF. It shows the rate at which the probability accumulates at each point. Given our CDF \(F_X(x)\), we calculate the PDF by finding the derivative where the CDF is differentiable. In cases where the CDF has discontinuities, the PDF will have corresponding spikes or jumps. Formally, if \(F_X(x)\) is continuous but not differentiable at some point (like at \(x = 1\) in our case), the PDF captures these peculiarities. For our function: \( F_X(x)\), the piecewise definition of CDF ensures that we can compute the corresponding piecewise PDF, noting particularly the behaviors around the discontinuity at \(x = 1\).