Does consistent estimators have in-variance property?

Question

If $(T_n)$ is a sequence of consistent estimators of a parameter $\theta$ ( i.e. for every $ \epsilon >0$ , $\lim_{n \to \infty} P [ \space |T_n -\theta|< \epsilon ]=1$ ) , then is it true that for any continuous function $f$ , $f(T_n)$ is a sequence of consistent estimators of $f(\theta)$ ?

Answer 1

$$\forallϵ\gt0,\quad\exists\alpha\gt0,\quad[|T_n−θ|<\alpha]\subseteq[|f(T_n)−f(θ)|<ϵ]$$