[Question] Limit and Integration

[Curl]

Suppose f(x,y) is some continuous, smooth function with the property
$$\lim_{y \to 0}f(x,y)=1$$
and g(x) is some other continuous smooth function.
I want to know if this is true:
$$\lim_{y \to 0} \int_{a}^{b}g(x)f(x,y)dx =? \int_{a}^{b}g(x)dx$$

How can I show that it is or isn't true? For which case will it be true or not true?

Thanks

 

[micromass]

Check out Lebesgue's monotone convergence theorem: http://en.wikipedia.org/wiki/Monotone_convergence_theorem (in the middle of the wiki page)

and his dominated convergence theorem: http://en.wikipedia.org/wiki/Dominated_convergence_theorem

 

[Curl]

Oh this is actually easy to see if you think of the integral as a Riemann sum.
I don't know why I always think of the solution right after I post the question.

So this is true in general, correct?

 

[micromass]

No, it certainly is not true in general. And I don't really know how Riemann sums help you here.

The problem is that you want to interchange two limits, this is not always allowed.