Let |f(x)|≤x2,∀x∈R Now, at x=0,|f(0)|≤0 ⇒f(0)=0 ∴f′(0)=
lim
h→0
f(h)−f(0)
h−0
=
lim
h→0
f(h)
h
.....(1) Now, |
f(h)
h
|≤|h|(∵|f(x)|≤x2) ⇒−|h|≤
f(h)
h
≤|h| ⇒
lim
h→0
f(h)
h
→0 .....(2) (using sandwich Theorem) ∴ from (1) and (2), we get f'(0) = 0, i.e. – f (x) is differentiable, at x = 0 Since, differentiability ⇒ Continutity ∴|f(x)|≤x2, for all x∈R is continuous as well as differentiable at x=0