Let f:R→R be a function such that |f(x)| ≤ x², for all x∈R. Then, at x=0, f is:

We have, −|f(x)| ≤ f(x) ≤ |f(x)|
Let us consider the continuity of the function at x=0.
We have, limx→0 −|f(x)| = 0 and limx→0 |f(x)| = 0.
Hence, by applying the sandwich theorem, limx→0 f(x) = 0
Hence, the function is continuous at x=0.
Now let us consider the differentiability of f(x) at x=0.
We have, limx→0 [f(x) - f(0)] / (x-0) = limx→0 f(x)/x
Since |f(x)| ≤ x², we have |f(x)/x| ≤ |x|
Therefore, limx→0 |f(x)/x| ≤ limx→0 |x| = 0
Hence, limx→0 f(x)/x = 0
Thus, f'(0) = 0, which means that f(x) is differentiable at x=0.
Therefore, f(x) is continuous as well as differentiable at x=0.