Given a $C^0$ function $g:[a,b]\to \mathbb{R}$ that is smooth everywhere except at $c$ (where $a<c<b$), and has positive derivative everywhere except at $c$, the claim is that there exists a function $G:[a,b] \to \mathbb{R}$ that is smooth and has positive derivative everywhere (including $c$), and that agrees with $g$ in some neighborhood about $a$ and some neighborhood about $b$.
According to Guillemin and Pollack's Differential Topology, one such function that fits the bill is $G(x) = g(a) + \int_{a}^{x}[k\rho(s)+g'(s)(1-\rho(s))]ds$, where $\rho$ is a smooth nonnegative function on $[a,b]$ that vanishes outside of $[u,v]$ (where $a<u<c<v<b$), is equal to $1$ on some neighborhood about $c$, and satisfies $\int_{a}^{b}\rho(s) ds = 1$. $k$ is the constant defined $k=g(b)-g(a)-\int_{a}^{b}g'(s)(1-\rho(s))ds$.
I have been able to show that $G$ as defined above agrees with $g$ on the required neighborhoods, and that $k>0$; however, I have been completely unable to prove that $G$ has positive derivative everywhere. Its derivative is certainly positive in the neighborhood about $c$, and near $a$ and $b$, but I don't see how one can prove that it is positive everywhere. I would greatly appreciate any help.
You are close to the answer. So I give a hint here.
Actually you are right. The hint is not depended on the condition $\int_a^b\rho dx=1$. And we should construct a $\rho$ satisfying the hint at first.
This Lemma can be found in Lee's Introduction to Smooth Manifolds.
Then choose $M=(a,b)$ and $F=[u,v]$, $G=(a,b)$. Then $\rho$ will satisfies the hint.