• # question_answer If $f\left( x \right)=\cos \pi \left( \left| x \right|+\left[ x \right] \right),$ then choose the incorrect option. A) $f$is continuous at $x=\frac{1}{2}$ B) $f$is continuous at $x=0$ C) $f$is differentiable in $\left( -1,0 \right)$ D) $f$is differentiable in $\left( 0,1 \right)$

Solution :

 $f(x)=\cos \pi (\left| x \right|+[x])$ $f\left( 0 \right)=1,f\left( 0-0 \right)=\cos \pi \left( -1 \right)=-1$ $f\left( \frac{1}{2} \right)=\cos \pi \left( \frac{1}{2} \right)=0,$
 $f\left( \frac{1}{2}-0 \right)=\cos \pi \left( \frac{1}{2} \right)=0$ $f\left( \frac{1}{2}+0 \right)=\cos \frac{\pi }{2}=0$ for $x\in \left( 0,1 \right),f\left( x \right)=\cos \pi x$ Which is differentiable for $x\in \left( -1,0 \right),$$f\left( x \right)=\cos \pi \left( -\pi -1 \right)=-\cos \pi x$ Which is differentiable

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