Given that $f$ is integrable on $[a,b]$ we have a partition $P=\{x_0, x_1, \cdots x_n\}$ such that for a given very small $\epsilon \gt 0$
$$
U(f,P) - L(f,P) \lt \epsilon
$$
Now, let's say in the interval $[x_{i-1}, x_i]$ at some $x$ we change the value of $f$ from $f(x)$ to $\alpha$. To prove that $f$ is still integrable, make the partition $P'$ such that
$$
P'= \{x_0 , x_1 \cdots x_{i-1}, x , x_i, \cdots x_n\}
$$
(a new point $x$, where we changed the value of $f$, is introduced in the partition points)