Let \(f\left(x\right)=\lim _{\theta \to 0}\left(\frac{\cos \pi x−{x}^{\left(\frac{2}{\theta }\right)}\sin \left(x−1\righ…

Let \(f\left(x\right)=\lim _{\theta \to 0}\left(\frac{\cos \pi x−{x}^{\left(\frac{2}{\theta }\right)}\sin \left(x−1\right)}{1+{x}^{\left(\frac{2}{\theta }\right)}\left(x−1\right)}\right),x\in R.\)

Consider the following two statements:

(I) \(f\left(x\right)\) is discontinuous at \(x = 1\).

(II) \(f\left(x\right)\) is continuous at \(x = – 1\).

Then,