Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:(= 2x^2 - 2x + 5",", "for" 0 ≤ x ≤ 2),(= (1 - 3x - x^2)/(1 - x) "," , "for" 2 < x < 4),(= (x^2 - 25)/(x - 5)",", "for" 4 ≤ x ≤ 7 and x ≠ 5),(= 7",", "for" x = 5):}`
Solution
The domain of f is [0, 7]
For 0 ≤ x ≤ 2, f(x) = 2x2 – 2x + 5, being a polynomial function is continuous.
For 2 < x < 4, f(x) = `(1 - 3x - x^2)/(1 - x)`, being a rational function is continuous except at the point where its denominator 1 – x = 0, i.e., at the point x = 1 which does not belong to (2, 4).
For 4 ≤ x ≤ 7, x ≠ 5, f(x) = `(x^2 - 25)/(x - 5)`, being a rational function is continuous except at the point where its denominator x – 5 = 0, i.e., at the point x = 5.
Continuity at x = 5
f(5) = 7 ...(Given) ...(1)
`lim_(x -> 5) "f"(x) = lim_(x -> 5) (x^2 - 25)/(x - 5)`
= `lim_(x -> 5) ((x - 5)(x + 5))/(x - 5)`
= `lim_(x -> 5) (x + 5)` ...[∵ x → 5, x ≠ 5 ∴ x – 5 ≠ 0]
= 5 + 5
= 10 ...(2)
From (1) and (2),
`lim_(x -> 5) "f"(x) ≠ "f"(5)`
∴ f is not continuous at x = 5.
Hence, f is continuous on its domain [0, 7] except at the point x = 5, where it is discontinuous.
