Find the points of discontinuity of the function f, where, (i) f(x) = {(4x + 5, if x ≤ 3), (4x - 5, if x > 3)

Question

Find the points of discontinuity of the function f, where, 

(i) \( f(x) = \begin{cases} 4x+5, & \quad \text{if } x ≤3\text{ }\\ 4x-5, & \quad \text{if } x>3 \text{ } \end{cases} \)

(ii) \( f(x) = \begin{cases} x+2, & \quad \text{if } x ≥2\text{ }\\ x^2, & \quad \text{if } x<2 \text{ } \end{cases} \)

(iii) \( f(x) = \begin{cases} x^3-3, & \quad \text{if } x ≤2\text{ }\\ x^2+1, & \quad \text{if } x>2 \text{ } \end{cases} \)

(iv) \( f(x) = \begin{cases} sinx, & \quad \text{} 0 ≤ x ≤ \frac{\pi}{4}\text{ }\\ cosx, & \quad \text{} \frac{\pi}{4}<x>\frac{\pi}{2} \text{ } \end{cases} \)

Answer 1

(i) f(3) = 12 + 5 = 17

∴ f(x) is discontinuous at x = 3

(ii) f(x) = 4

∴ f(x) is continuous for all x ∈ R

(iii) f(x) = 8 – 3 = 5

∴ f(x) is continuous for all x ∈ R

(iv) \( f(x) = \begin{cases} sinx, & \quad \text{} 0 ≤ x ≤ \frac{\pi}{4}\text{ }\\ cosx, & \quad \text{} \frac{\pi}{4}<x>\frac{\pi}{2} \text{ } \end{cases} \)

∴ f(x) is continuous for all x ∈ [0, π/2]