Question

Show that f(x) = x² is continuous and differentiable at x = 0

Answer 1

R f'(0) = `lim_("h" -> 0^+) ("f"(0 + "h") - "f"(0))/"h"`

= `lim_("h" -> 0) ("h"^2 - 0)/"h"`     ...[∵ f(x) = x²]

= `lim_("h" -> 0) "h"`   ...[∵ h → 0 ∴ h ≠ 0]

= 0

Similarly, it can be shown that L f'(0) = 0

∴ R f'(0) = L f'(0) = 0

∴ f is differentiable at x = 0

and hence continuous at x = 0.