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I want to show that for $\gamma: [a,b] \subseteq \mathbb{R} \rightarrow V$ continously differentiable where V is a bounded subset of $\mathbb{R}^2$. There is always a sequence of functions $f_n:\mathbb{R} \rightarrow \mathbb{R}$ that approximates this curve if we assume that for $\gamma(t)=(x_1(t),x_2(t))$ we have $\dot{x}(t)\ge0$. Is there a theorem that could help me with this?

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Depends on what you mean by approximating? I could use $\gamma$ to approximate $\gamma$, but that's quite lame. What properties do you want your approximation to satisfy? Also, it doesn't make sense to ask that the derivative is positive considering it has to be a vector. –  Patrick Da Silva Mar 15 '13 at 19:04
    
I can approximate any continuous function by an infinity differentiable one (in the sup-norm). –  Baby Dragon Mar 15 '13 at 19:10
    
what is the name of your theorem? did you keep in mind that i have a curve in two dimensional space, but want to approximate it by an actual function mapping from $\mathbb{R}$ to $\mathbb{R}$? –  Xin Wang Mar 15 '13 at 19:13
    
Ok I see, you condition on $\dot{x}$ makes it look like $x$ is almost invertible. The issue is the set of $t$-values where $\dot{x}=0$. Supposing we assume that $\dot(x)$ is continuous, we have two cases. The first is that a point, $\dot{x}(t)$ is zero in some closed interval $[c,d]$. The second case is that the first case does not happen. In the first case $x$ is constant over the interval, so the curve $\gamma$ just jumps up and down for a while. Elsewhere $x$ is actually invertible, so just invert it. –  Baby Dragon Mar 15 '13 at 19:47

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