Evaluate a Piecewise Linear Function

Problem Statement

You are given a list of 2D points points representing a piecewise linear function. The list contains n points (x_i, y_i) where x_0 < x_1 < ... < x_{n-1}. Your task is to implement a function that evaluates this function at any given point x.

If x falls exactly on one of the x_i coordinates, return the corresponding y_i. If x lies between x_i and x_{i+1}, return the result of the linear interpolation between the two points. If x is outside the range [x_0, x_{n-1}], return null (or the equivalent None/Empty value).

Function Signature

def evaluate_piecewise(points, x):

Constraints

• points: A list of tuples (x, y) sorted by x.
• n (number of points): 2 <= n <= 10^5.
• x: A floating-point number.
• Time complexity should be better than O(n).

Examples

Example 1:

• Input: points = [(0, 0), (2, 4), (4, 4)], x = 1
• Output: 2.0
• Explanation: Since 1 is between 0 and 2, the line segment between (0,0) and (2,4) has a slope of 2. y = 0 + 2 * (1 - 0) = 2.

Example 2:

• Input: points = [(0, 0), (2, 4), (4, 4)], x = 5
• Output: null

Edge Cases

• x is exactly the first or last point.
• x is significantly outside the bounds.
• The function is horizontal (slope = 0) between two points.