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