Consider the piecewise-defined function b(x)= cases ( 1 x ), & x<0

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the piecewise-defined function b(x)={sin⁡(1x),x<0,x2 ln⁡(x+1),0≤x<3,9−x,x≥3.b(x)=\begin{cases} \sin\left(\dfrac{1}{x}\right), & x<0,\\ x^{2}\,\ln(x+1), & 0\le x<3,\\ \sqrt{9-x}, & x\ge 3. \end{cases} Define h(x)=ln⁡∣b(x)∣h(x)=\ln\bigl|b(x)\bigr|. (1) Part 1: (a) For each of the three formula pieces that define bb, determine the subset of R\mathbb{R} on which that piece is continuous. Provide explicit interval notation and brief justification for each piece. (2) Part 2: (b) Identify every point at which bb fails to be continuous. For each such point, classify the discontinuity as removable, jump, or essential (infinite/oscillatory), and give a supporting argument based on limits. (3) Part 3: (c) Determine precisely the set of all real numbers on which hh is continuous. In your answer, discuss how the absolute-value and natural-logarithm operations affect the continuity—or discontinuity—of bb. (4) Part 4: (d) State whether hh can be extended to a new function that is continuous at any of the points where it is currently discontinuous. If an extension is possible, supply an explicit definition on the additional point(s) and justify continuity of the extended function there.

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