A function defined by f(x) = x + a, for x < 0 = x, for 0 ≤ x < 1 = b - x, for x ≥ 1

Question

i. A function defined by

f(x) = x + a, for x < 0

= x, for 0 ≤ x < 1

= b - x, for x ≥ 1

is continuous in [-2, 2]. Show that (a + b) is even.

ii. Show that \(f(x) = \begin{cases} \frac{sin\,x}{x}, & x<0\\ x+1, & x \geq 0 \end{cases}\) is a continuous function.

Answer 1

i. f(x) is continuous in [- 2, 2].

∴ f(x) is continuous at x = 0.

ii. f(0) = (0) + 1 = 1

Hence, f(x) is continuous at x = 0.