Applying the IVT to prove a value is attained

Since $h(x) = 3x - \sin(\pi x)$ is a difference of a polynomial and a sine function, both of which are continuous everywhere, $h$ is continuous on all of $\mathbb{R}$, and in particular on the closed interval $[0,1]$. This continuity is exactly the hypothesis the IVT requires.

Evaluate the endpoints: $h(0) = 3(0) - \sin(0) = 0$, and $h(1) = 3(1) - \sin(\pi) = 3 - 0 = 3$. Since $h(0) = 0 < 1 < 3 = h(1)$, the value $N=1$ lies strictly between $h(0)$ and $h(1)$.

Because $h$ is continuous on $[0,1]$ and $1$ is between $h(0)$ and $h(1)$, the Intermediate Value Theorem guarantees there exists at least one $c \in (0,1)$ with $h(c) = 1$. This proves the desired value is attained, without needing to solve for $c$ explicitly.