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The function $f(x)=\cos ^{-1}(\cos x)$, is |
discontinuous at infinitely many-points everywhere differentiable such that f'(x) = 1 not differentiable at $x=n \pi, n \in Z$ and $f'(x)=1, x \neq n \pi$ not differentiable at $x=n \pi, n \in Z$ and $f'(x)=(-1)^n, x \in(n \pi,(n+1) \pi), n \in Z$ |
not differentiable at $x=n \pi, n \in Z$ and $f'(x)=(-1)^n, x \in(n \pi,(n+1) \pi), n \in Z$ |
The graph of $f(x)=\cos ^{-1}(\cos x)$ as given in Figure. It is evident from the curve y = f(x) that f(x) is everywhere continuous but it is not differentiable at $x=n \pi, n \in Z$ Also, $f'(x)=(-1)^n, x \in(n \pi,(n+1) \pi)$, where $n \in Z$ We also observe that f(x) is an even periodic function with period $2 \pi$.
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