Let $\omega_t$ be a differential form. Prove that $\varphi(t)=\int_\lambda\omega_t$ is continuous function.

Question

For each $t\in T\subset\mathbb{R}^m$, let $w=\sum_{i=1}^na_i(t,x)dx_i$ a differential form in a open set $U\subset \mathbb{R}^n$, such that $a_i:T\times U\to\mathbb{R}$ are continuous functions. For each rectifiable path $\lambda:[a,b]\to U$, prove that $\varphi:T\to\mathbb{R}$ defined by $\varphi(t)=\int_\lambda\omega_t$ is continuous.
I was writing the definition of continuous functions to get some idea but i get nothing less this $$|\varphi(t)-\varphi(t_0)|=|\int_\lambda\omega_t-\int_\lambda\omega_{t_0}|=|\int_\lambda\sum_{i=1}^na_i(t,x)dx_i-\int_\lambda\sum_{i=1}^na_i(t_0,x)dx_i|$$ And after that tried to think in how could this gonna be $<\epsilon$. Anyway, i'm stuck on that question and need a help.