Let $M, N$ be compact, connected, oriented manifolds. The degree of a map $f:M \rightarrow N$ is defined as the integer $k$ which satisfies $\int_{M} f^{*}\omega = k\int_{N}\omega$. Using the fact that homotopic maps, say $f, g$ induce the same map on the De Rham cohomology, we have $\int_{M} f^{*}\omega - \int_{M} g^{*}\omega = \int_{} d\nu$ for $\nu$ some $n-1$ form. Now if the manifold has empty boundary, we can use Stokes's theorem to conclude that this equals 0.
However, what about when the manifold does not have empty boundary? A proof that I found relies on the fact that if you have a homotopy $H(x,t)$ then $t \mapsto H_{t}^{*}\omega$ is continuous, and then using the continuity of the integral, concluding that (since the degree is always an integer), $k(t)=\frac{\int_{M}H_{t}^{*}\omega}{\int_{N}\omega}$is also continuous, hence constant. Is there any other way of proving this?