Counterexample/Example - continuity of parametric integral

Question

Let's consider a parametric integral $F(x):=\int\limits_a^{b}K(x,t) dt$ where $K:\mathbb{R}^2\to \mathbb{R}$ and $a,b \in \mathbb{R}$. The function $K$ is Riemann-integrable.

We have the statement that if $K$ is continuous then the parametric integral $F(x)$ is also continuous.

So if $K$ is not continuous I can't use this statement. However, I was wondering if there exists an example of a discontinuous $K$ and a continuous parametric integral $F(x)$.

Answer 1

Try $a=0,b=1$ and $K=1_{[0,x]}(t)$ which is not continuous (but $F(x)=\int_0^{\min(1,x)} 1\,\mathrm dt=\min(1,x)$ is continuous).

Answer 2

If $K$ does not depend on $x$, then $F$ is constant. But $K$ might still be discontinuous as a function of $t$.