Limits, Continuity & Differentiability
Continuity of Composite Functions
Grade 12
Question:
<p>If <span style='display:inline-block'>f: \mathbb{R} \to \mathbb{R}\)</span> is a function defined by <span style='display:inline-block'>f(x) = [x]\cos\left(\frac{2x-1}{2}\pi\right)\)</span>, where <span style='display:inline-block'>[x]\)</span> denotes the greatest integer function, then f is</p>
<p>(a) continuous for every real x</p>
<p>(b) discontinuous only at x = 0</p>
<p>(c) discontinuous only at non-zero integral values of x</p>
<p>(d) continuous only at x = 0</p>
Step-by-Step Solution
Key Concept: The product of a discontinuous function (at integers) and a continuous function is discontinuous where the first function is discontinuous.
<p>The greatest integer function [x] is discontinuous at every integer. The cosine term is continuous everywhere. The product f(x) = [x]·cos(...) is discontinuous wherever [x] is discontinuous, which occurs at all non-zero integral values of x. At x = 0, [0] = 0, so f(0) = 0·cos(...) = 0, and the limit also equals 0, making f continuous at x = 0.</p>
Correct Answer: C