Proving Continuity of a Lebesgue Integrable product of functions

Question

I was doing some practice Measure Theory questions and came across a large set of questions that are very similar to the following:

Let $g \in L^1([0,1])$ and define $f$ such that $\forall a\in \mathbb{R}$ by $$ f(a) = \int_{[0,1]} g(x) \cos^5(x^4a)\ dx, $$ as a Lebesgue integral with respect to $x$ for each fixed $a\in \mathbb{R}$. Prove $f$ is continuous.

I am still fairly new to this stuff, and after a few attempts, I am at a loss with how to arrive at $f$ being continuous.

I tried the MCT and Lebesgue Dominated Convergence Theorem, but there isn't any subsequence that I could use (at least I don't think so). I also tried to use H$\ddot{\text{o}}$lder's Inequality but only came up with that $f(a) \leq |f(a)| \leq \int_{[0,1]} |g(x)| $ since $|cos^5(x^4a)| \in [0,1],\ \forall x,a \in \mathbb{R} $. Any help would be appreciated!

EDIT: Fixed a typo with variables (was a $y$ that was supposed to be $a$).

Answer 1

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence in $\mathbb{R}$ that converges to a point $a$. You want to show that $f(a_n) \to f(a)$. Indeed, define the function: $$h(x,y) = g(x)\cos^5(x^4 y)$$ This function is measurable and obviously, $|h(x,y)| \leq g(x)$. Observe that $h(x,a_n) \to h(x,a)$ for every $x$ due to the continuity of $\cos^5$. Now, define a sequence of functions: $$f_n(x) := g(x) \cos^5(x^4 a_n)$$

This sequence of functions point-wise converges to $h(x,a)$ and is dominated by $g(x)$. By the Dominated Convergence Theorem, you have that: $$\lim_{n \to \infty} f(a_n) = \lim_{n \to \infty} \int_{[0,1]} f_n(x) \ dx = \int_{[0,1]} h(x,a) \ dx = f(a)$$

I'm a bit surprised that you haven't proved this or have been shown the proof of a similar result; this is the sort of thing that's immediately proved right after the proof of the DCT is given. It's supposed to display just how useful the DCT actually is. You should probably check your book and see if the author actually proved something quite similar to this.

Answer 2

Start with a definition of continuity you are comfortable with. I will use one in terms of sequences like you mentioned in your post.

Let $a\in \mathbb R$ be given, and consider a sequence $a_n\to a$. If we show $f(a_n)\to f(a)$, then we will have proved $f$ is continuous at $a$. Start by writing $$ f(a_n) = \int g(x)\cos^5(x^4a_n)\,dx. $$ Now it is natural at this point to consider the sequence of functions $F_n(x):= g(x)\cos^5(x^4 a_n)$, as well as the function $F_a(x):= g(x)\cos^5(x^4a)$. I'll leave you to think about whether and how the dominated convergence theorem applies here to let you conclude $f(a_n)\to f(a)$.