I'm trying to prove that if $f,g:[a,b] \rightarrow \mathbb{R}$ are continuous and defined on $[a,b]$, $f(a)=g(a), f(b)=g(b)$ and $H$ is a homotopy such that $H(x,0) = g(x)$ and $H(x,1) = f(x)$ then $\int_{a}^{b}f(x)dx = \lim_{t\rightarrow 1} \int_{a}^{b} H(x,t)dx$.
So what I have so far is the following:
Let $m = sup_{x\in [a,b]} |f(x) - g(x)|$, then $\int_{a}^{b}mdx \ge | \int_{a}^{b}f(x) - g(x)dx|$ for all $x\in [a,b]$. So I need to prove there exists a delta such that for every $\epsilon \gt 0$ there exists a $\delta$ such that $|t-1| \lt \delta$ implies that $|int_{a}^{b} f(x) - H(x,t)dx| \lt \epsilon$.
I'm having trouble figuring out how to define my delta. I know it has to do with $m$ but I can't figure how to apply to the$|t-1| \lt \delta$.