A function between two others?

Does there exist a function $g(x)$ such that for all (or for all positive) x-values

$c_1 \sqrt{x} \leq g(x) \leq c_2 \sqrt{x}, c_1 < c2$

but $g(x) \neq c_3 \sqrt{x}$ ?

Or, in general, does there exist a function $g(x)$ for some other function $f(x)$ such that for all (or for all positive) x-values

$c_1 f(x) \leq g(x) \leq c_2 f(x), c_1 < c2$

but $g(x) \neq c_3 f(x)$ ?

Note: $c_1, c_2, c_3 \in \mathbb{R}$

Edit: Obviously, $g(x) = c_1 \sqrt{x} + h(x)$ with $0 \leq h(x) \leq (c_2 - c_1) \sqrt{x}$ works, so there are many solutions. What I meant, was that $g(x)$ should be independent of $\sqrt{x}$ , so $\sqrt{x}$ shouldn't appear in the definition of $g(x)$ , maybe there is also formal definition to independent.

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
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