Proving continuity of an integral I have the following function:
$$I_n(a)=\int_{-\infty}^{\infty}x^6e^{-x^2}\operatorname{sech}^n(ax)dx$$
where $\operatorname{sech}(x)=\frac{2}{e^x+e^{-x}}$ is the hyperbolic secant.
Clearly, the integral is a beast to evaluate.  However, all I need is to prove the continuity of $I_n(a)$ at $n=2,4,6$; I don't need the closed form of $I_n(a)$.  In particular, I am interested in the continuity of $I_n(a)$ in the neighborhood of $a$ around zero, i.e. $a\in[-\epsilon,\epsilon]$ for some small positive $\epsilon$ (I plotted it in that region and it "looks" continuous to me, however, I'd like a rigorous proof).  
This seems like a simple problem, but I have no idea where to start.  Someone suggested the Lebesgue Dominated Convergence Theorem, but unfortunately, I am shaky on the measure theory, having never taken a course on the subject.  I read the wikipedia page, but have no idea how to use.  Is there perhaps a simpler way?  If not, can someone help?
 A: The Dominated Convergence Theorem has been explained to me, and I think I can use it as follows to prove the continuity of $I_n(a)$.
Let $g(x)=x^6e^{-x^2}$.  Clearly $g(x)$ is integrable, as $\int_{-\infty}^{\infty}|g(x)|dx$ is the sixth moment of the Gaussian distribution times a constant.  Since $0<\operatorname{sech}(x)\leq1$, the integrand is dominated by $g(x)$, i.e. $|x^6e^{-x^2}\operatorname{sech}^n(ax)|\leq g(x)$.
To prove continuity, we can show that $\lim_{a\rightarrow a_0} I_n(a)=I_n(a_0)$ for the region $a_0\in[-\epsilon,\epsilon]$ in which we are interested.  We can show this as follows:
$$\begin{array}{rcl}\lim_{a\rightarrow a_0}I_n(a)&=&\lim_{a\rightarrow a_0}\int_{-\infty}^{\infty}x^6e^{-x^2}\operatorname{sech}^n(ax)dx\\
&=&\int_{-\infty}^{\infty}x^6e^{-x^2}\lim_{a\rightarrow a_0}\operatorname{sech}^n(ax)dx\\
&=&\int_{-\infty}^{\infty}x^6e^{-x^2}\operatorname{sech}^n(a_0x)dx\\
&=&I_n(a_0)
\end{array}$$
where the movement of limit inside the integral in the second equality is allowed since we have shown that the conditions for the Dominated Convergence Theorem hold.  Thus, we proved that $I_n(a)$ is continuous on the entire real number domain for $n\geq 1$.
