List of top Mathematics Questions on Continuity and differentiability asked in CPET

For a function \(f: \mathbb{R}\to\mathbb{R}\), consider the following statements:
(I) If \(\lim_{h\to0}\{f(x+h)-f(x-h)\} = 0\), then f is continuous on \(\mathbb{R}\).
(II) If \(\lim_{h\to0}\dfrac{f(x+h)-f(x-h)}{2h} = 0\), then f is differentiable on \(\mathbb{R}\).
Choose the correct answer.

Which one of the following options is true for the function \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x) = \min\{|x|,\, x^2-1\}\), \(x\in\mathbb{R}\)?