Continuity and DifferentiabilityHard

Question

If f(x) = x - 1, then on the interval [0, π]

Options

A.tan [f(x)] and 1/f(x) are both continuous
B.tan [f(x)] and 1/ f(x) are both discontinuous
C.tan [f(x)] and f-1(x) are both continuous
D.tan [f(x)] is continuous but 1/ f(x) is not continuous

Solution

Given, f(x) = x - 1 for 0 ≤ x ≤ π
∴     [f(x)] =
⇒     tan [f(x)] =
∴     tan [f(x)] = - tan 1
and tan[f(x)] = 0
So, tan f(x) is not continuous at x = 2
Now,     f(x) = x - 1
⇒     f(x) =
⇒    
Clearly, 1/ f(x) is not continuous at x = 2
So, tan [f(x)] and tan are both discontinuous at x = 2

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