prove the integral function is continuous 
Let $f: [a,b]\times [c,d]\to \mathbb{R}, (x,\alpha)\to f(x,\alpha)$ is a 2 variable continuous function on $[a,b]\times [c,d]$, then $h(\alpha )=\int _a^b f(x, \alpha )dx$ is a continuous function on $[c,d]$.

My attempt: I've tried to prove this using the definition of 2-var continuous (that is, for any $(x, \alpha)$, $\epsilon >0$, $\exists \delta$, all $(y, \beta)$ satisfies $\sqrt{(y-x)^2+(\beta-\alpha)^2}<\delta$ yields $|f(y, \beta)-f(x, \alpha)|<\epsilon$) but since $\epsilon$ depends on $x$ so I can't go directly from here.Thanks for any help.

Hint: the projection of a continuous function onto any coordinate is continuous, and the integral of a continuous real function is continuous.

In a general sense, we may define the space $\mathbb{R}^2$ to be the collection of ordered pairs from $\mathbb{R}$, equipped with the property that the projection of any continuous function onto either coordinate is always continuous. This is one possible way to define the product topology on $\mathbb{R}^2$ (which happens to be just the usual Euclidean topology).
