Question

Discuss the continuity and differentiability of f(x) = x |x| at x = 0

Answer 1

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.