First slide

Continuity

Question

The set of points of continuity of the function $f (x)$ $= \sqrt{\frac{1}{2} - \cos^{2} x}$ is:

Moderate

A

${x : \frac{π}{4} + 2 n π \leq x \leq \frac{3 π}{4} + 2 n π, n \in I}$
B

${x : \frac{5 π}{4} + 2 n π \leq x \leq \frac{7 π}{4} + 2 n π, n \in I}$
C

${x : \frac{π}{4} + 2 n π \leq x \leq \frac{3 π}{4} + 2 n π} \cup {x : \frac{5 π}{4} + 2 n π \leq x \leq \frac{7 π}{4} + 2 n π}$
D

None of these

Solution

The function $f (x)$ is continuous at all points where $\frac{1}{2} - \cos^{2} x \geq 0 \Rightarrow | \cos x | \leq \frac{1}{\sqrt{2}}$
$\Rightarrow \frac{π}{4} + 2 n π \leq x \leq \frac{3 π}{4} + 2 n π$

Or $\Rightarrow \frac{5 π}{4} + 2 n π \leq x \leq \frac{7 π}{4} + 2 n π, n \in I$

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