Continuous function $F(t)=\int_a^bf(x)cos(tx)dx$

Question

I was wondering if the next problem is a particular case of this exercise:

If $f$ is continuous on rectangle $[a,b]\times[c,d]$ and if $g$ is Riemann integrable on $[a,b],$ then the function $F$ defined by the equation $$F(y)=\int_a^bg(x)f(x,y)dx$$ is continuous on $[c,d]$.

And this is the problem:

Let $f$ be a continuous function defined on $[0,\pi/2]$. Prove that function $F:[0,\pi/2]\to \mathbb R$ defined by $$F(t)=\int_a^bf(x)cos(tx)dx$$ is continuous.

Answer 1

In your problem, $f$ is continuous and hence Riemann integrable, and $g(x,t)=\cos(xt)$ is continuous, so you can indeed apply the proposition you mentioned, which proves that the function $F$ in the problem is continuous.

To see that $g(x,t)=\cos(xt)$ is continuous, note that it is a composition $g = \sin\circ \,h$ of two continuous functions, where $h(x,t)=xt$.