Let f(x)={sgn⁡(x)+x;−∞<x<0−1+sin⁡x;0≤x<π2cos⁡x;π2≤x<∞f(x) = \begin{cases} \operatorname{sgn}(x) + x; & -\infty < x < 0 \\ -1 + \sin x; & 0 \le x < \frac{\pi}{2} \\ \cos x; & \frac{\pi}{2} \le x < \infty \end{cases}…

Mathematics

Problem

Let f(x)={sgn⁡(x)+x;−∞<x<0−1+sin⁡x;0≤x<π2cos⁡x;π2≤x<∞f(x) = \begin{cases} \operatorname{sgn}(x) + x; & -\infty < x < 0 \\ -1 + \sin x; & 0 \le x < \frac{\pi}{2} \\ \cos x; & \frac{\pi}{2} \le x < \infty \end{cases}. Discuss the continuity & differentiability at x=0x = 0 & π2\frac{\pi}{2}.

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