Question

At the given points x₀ discover whether the given function is continuous or discontinuous citing the reasons for your answer.

(i) x₀ = 1, \( f(x) = \begin{cases} \frac{x^2-1}{x-1}, & \quad \text{ } x ≠ 1\text{ }\\ 2, & \quad \text{ } x=1 \text{ } \end{cases} \)

(ii)  x₀ = 3, \( f(x) = \begin{cases} \frac{x^2-9}{x-3}, & \quad \text{if } x ≠ 3\text{ }\\ 5, & \quad \text{if}\, x=3 \text{ } \end{cases} \)

Answer 1

(i) Given f(x₀) = 1

∴ f(x) is continuous at x₀ = 1

(ii)  x₀ = 3, \( f(x) = \begin{cases} \frac{x^2-9}{x-3}, & \quad \text{if } x ≠ 3\text{ }\\ 5, & \quad \text{if}\, x=3 \text{ } \end{cases} \)

a

∴ f(x) is not continuous at x₀ = 3