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Let f(x) = cos x for 0 <= x < pi/2 and f(x) = (pi/2 - x)² for pi/2 <= x < pi, extended periodically with period pi. Which of the following statements is INCORRECT?

1. The range of f is [0, pi²/4)
2. f is continuous for all real x, but not differentiable for some real x
3. f is continuous for all real x
4. The area bounded by y = f(x) and the x-axis from x = -n*pi to x = n*pi is 2*n*(1 + pi³/24) for a given n in N

Correct answer: f is continuous for all real x

Solution

At the period boundary the left limit is pi²/4 while the right value is cos 0 = 1; pi²/4 is about 2.47 not equal to 1, so f is NOT continuous everywhere. Thus 'f is continuous for all real x' is the incorrect statement.

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