I'm trying to show that the function
$v(t) = \int_0^1 e^{\sqrt{x^2+t^2}} d\lambda$
(with $\lambda$ being the Lebesgue measure) is continuous for all $t \in \mathbb{R}$. I've however hit a little snag.
I'm trying to show that $v(t)$ satisfies the continuity lemma, that is, that:
- $ x \mapsto e^{\sqrt{x^2+t^2}} \in \mathcal{L}^1([0,1],\lambda)$ for every fixed $t$
- $t \mapsto e^{\sqrt{x^2+t^2}}$ is continuous for every fixed $x$.
- $|e^{\sqrt{x^2+t^2}}| \leq w(x)$ for all $(t,x) \in \mathbb{R} \times [0,1]$ some $w(x) \in \mathcal{L}^1$
One and two I've done, and I'm pretty confident they're alright. It's number three that's giving me trouble - I'm completely stuck. Anyone got any tips/help?
Much appreciated! :-)