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Question
Discuss the continuity and differentiability of f(x) = x |x| at x = 0
Solution
f(x) = x |x|
f(x) = x(– x), x < 0
= x(x), x ≥ 0
Continuity at x = 0:
`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) (-x^2)` = 0
`lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) (x^2)` = 0
f(0) = 0
∴ `lim_(x -> 0^-) "f"(x) = lim_(x -> 0^+) (x)` = f(0)
∴ f(x) is continuous at x = 0
Differentiability at x = 0:
L f'(0) = `lim_("h" -> 0^-) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^-) (-"h"^2 - 0)/"h"`
= `lim_("h" -> 0^-) (- "h")` ...[∵ h → 0, ∵ h ≠ 0]
= 0
R f'(0) = `lim_("h" -> 0^+) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^+) ("h"^2 - 0)/"h"`
= `lim_("h" -> 0^+) ("h")` ...[∵ h → 0, ∵ h ≠ 0]
= 0
∴ L f'(0) = R f'(0)
∴ f(x) is differentiable at x = 0.
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