Equicontinuous, differentiable continuous problem

Question

Assume that each of {$f_n : [0, 1] \rightarrow R$} is continuously differentiable

I know that if {$f_n'$} is uniformly bounded, {$f_n$} is equicontinuous.

However, the converse is NOT true.

I want to find this example to show that the converse is NOT true

Answer 1

The reason this is not true is because equicontinuity does not restrict the same variation over the whole domain. The problem is essentially the same as showing that a continuous function need not be uniformly continuous if the domain is not compact. A concrete example is $f_n(x) = f(x) = \frac{1}{1-x}$ for all $n$. The family $\{f_n\}$ is equicontinuous on $[0, 1)$ because $f$ is continuous, but $f'$ is not bounded.

Answer 2

Hint: any uniformly convergent sequence of continuous functions on [0,1] is equicontinuous. Think of sine waves of decreasing amplitude and increasing frequency.