Continuous Functions

(a) A function representing a ratio of two continous functions will be (polynomials in this case) discontinuous only at points for which the denominator zero. But in this case

(x4 + 4x3 + 8x2 + 8x + 4) = (x2 + 2x + 2)2 = [(x + 1)2 + 1]2 > 0 (always greater than zero)

Hence f(x) is continuous throughout the entire real line.

(b) The function f(x) suffers discontinuities only at points for which the denominator is equal to zero i.e. 4 cos x - 2 = 0 or cos x = 1/2 ⇒ x = xn Thus the function f(x) is continuous everywhere, except at the point xn