Discontinuities in Business

Business Calculus · Axiom Academy

EXAMPLE Discontinuities in Business Identifying and analyzing jumps in business functions Left and right limits exist but are different. Common in tiered pricing. Limit exists but function value is undefined or different. Often a "hole" in the graph. Function approaches ±∞. Occurs when dividing by zero. Example 1: Shipping Cost Function A company charges for shipping based on package weight (in pounds): Identify any discontinuities and explain their business meaning. Left limit (approaching 5 from below): Right limit (approaching 5 from above): Since the left limit ( 17.50) ≠ right limit ( 20), this is a jump discontinuity . Shipping cost jumps from 17.50 to 20 at 5 lbs The discontinuity represents a pricing tier change. A customer with a 4.9 lb package pays 17.50, but adding 0.2 lbs jumps the cost to 20.05. This creates an incentive to split shipments or reduce package weight. Example 2: Tax Bracket Function A simplified income tax function (in thousands of dollars): Is this function continuous? Where do discontinuities occur? Both equal 5,000, so the function is continuous at x = 50 . Both equal 17,500, so the function is continuous at x = 100 . This tax function is continuous everywhere — it's designed so there's no sudden jump when crossing bracket thresholds. This is how real progressive tax systems work: only income above each threshold is taxed at the higher rate. Example 3: Average Cost Discontinuity A company's average cost function is:

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