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Question
Discuss the continuity of f on its domain, where f(x) `{:(= |x + 1|",", "for" -3 ≤ x ≤ 2),(= |x - 5|",", "for" 2 < x ≤ 7):}`.
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Solution
The domain of f is [– 3, 7].
∵ |x + a| is continuous for all x ∈ R,
We need to consider the continuity of f only at x = 2
f(x) = |x + 1|, for – 3 ≤ x ≤ 2
∴ f(2) = |2 + 1|
= 3
`lim_(x -> 2^-) f(x) = lim_(x -> 2) |x + 1|`
= |2 + 1|
= 3
Also, f(x) = |x – 5|, for 2 < x ≤ 7
∴ `lim_(x -> 2^+) f(x) = lim_(x -> 2) |x - 5|`
= |2 – 5|
= 3
∴ f(2) = `lim_(x -> 2^+) f(x) = lim_(x -> 2^-) f(x)`
∴ f is continuous at x = 2.
Hence, f is continuous on its domain [– 3, 7].
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