I have a question about extending the interval of an estimate using continuity.
So suppose that I have positive constants $c_1, c_2, D$, some real number $r >1$ and continuous function $f(x) : [0,\infty) \rightarrow \mathbb{R} $ such that the following estimate holds for all $x \geq 1$
\begin{equation} c_1 \exp( -D x^\frac{2}{r}) \leq f(x) \leq c_2 \exp( -D x^\frac{2}{r}) \end{equation}
How can I extend the above result (with possibly different constants $c_{1}^{*}$ and $c_{2}^{*}$) to all $x>0$ using the function $x \rightarrow \exp( -D x^\frac{2}{r}) $ is bounded on $[0,1]$ above and below by positive constants depending only on $r$?
Will the same argument work if the origional estimate held only for $x > n$ where $n>2$?