Prove Continuity of a multivarible function. I'm trying to prove the following:

Let $f:\mathbb{R^n}\times\mathbb{R} \to \mathbb{R}$ be a continuous function. We define $$F(x,t) = \int_{0}^{t}f(x,s)ds  $$ Prove that F is also continuous.

I tried to use the limit definition of continuity but have not achieved anything worthed. Can you help me with this please?
 A: Sketch/Hint: Let me show you one "follow your nose" way to tinker until you reach a conclusion. We want to show that $|F(x_k,t_k) - F(x,t)| \to 0$ as $(x_k,t_k) \to (x,t)$ in $\mathbb R^n \times \mathbb R$. Since we know that $|f(x_k,s)-f(x,s)| \to 0$ as $k \to \infty$ (why?) and $f(x,s)$ is the integrand inside the integral definition $F(x,t)$, we can try to rewrite $|F(x_k,t_k) - F(x,t)|$ as an integral of a difference. The only real complication is that the integrals defining $F(x_k,t_k)$ and $F(x,t)$ have different bounds, but we get by this using elementary properties of the integral:
\begin{align*}
\left|F(x_k,t_k) - F(x,t)\right| &= \left|\int_0^{t_k}f(x_k,s)\,ds - \int_0^tf(x,s)\,ds\right|\\
    &= \left|\int_0^tf(x_k,s)-f(x,s)\,ds-\int_{t_k}^tf(x_k,s)\,ds\right|\\
 &\leq \int_0^t\left|f(x_k,s)-f(x,s)\right|\,ds + \left|F(x_k,t_k) - F(x_k,t)\right|
\end{align*}
The last expression is a sum of two things which both $\to 0$ as $(x_k,t_k) \to (x,t)$ by the continuity of $f$ (and the Fundamental Theorem of Calculus). I'll leave the details to you.
