Determine the continuity of a piecewise defined function at a point

Learning Objectives Define continuity at a point using the three-part definition. Evaluate the left-hand and right-hand limits of a piecewise function at a specific point. Determine the value of a piecewise function at a specific point. Systematically apply the three conditions of continuity to determine if a piecewise function is continuous at a given point. Identify and name the type of discontinuity (jump or removable) if a function is not continuous. Solve for an unknown constant that will make a piecewise function continuous at a specific point. Ever seen a badly edited movie where a character suddenly jumps from one spot to another? 🎬 Piecewise functions can do that too, and our job is to find those 'jumps'! This tutorial will guide you through the process...

TermDefinitionExample Piecewise Defined FunctionA function built from two or more different function 'pieces', where each piece applies to a different part of the domain.f(x) = { x^2, if x < 2; x + 2, if x ≥ 2 }. For any x-value less than 2, you use x^2. For any x-value greater than or equal to 2, you use x + 2. Continuity at a Point (c)A function is continuous at a point 'c' if its graph is unbroken at that point. You can draw the graph through the point without lifting your pencil.The function f(x) = x^2 is continuous at x = 3 because there is no hole, jump, or asymptote at that point. Left-Hand LimitThe value that a function f(x) approaches as x gets closer and closer to a point 'c' from the left side (i.e., through values less than c).For f(x) = { x, i...

The Three Conditions for Continuity A function f(x) is continuous at a point x = c if and only if all three of the following conditions are met: 1. f(c) is defined. 2. lim_{x→c} f(x) exists. 3. lim_{x→c} f(x) = f(c). This is the formal definition and the ultimate checklist for determining continuity. You must verify all three conditions. If any one of them fails, the function is discontinuous at x = c. Existence of a Limit The limit lim_{x→c} f(x) exists if and only if the left-hand and right-hand limits are equal: lim_{x→c⁻} f(x) = lim_{x→c⁺} f(x) = L. For piecewise functions, this is the key to satisfying the second condition of continuity. You must calculate the limit from both sides of the point in question and check if they match.