Confirming Continuity over an Interval Practice Test

Show $|x-3|$ is continuous on $[1,5]$ and adding a constant $k$ preserves continuity; endpoints are defined.

Show $p$ is continuous only at $x=3$, since absolute value can be discontinuous elsewhere.

Show $p$ is differentiable at $x=3$ for all $k$ and continuous elsewhere.

Show $\lim_{x\to 1}p(x)$ and $\lim_{x\to 5}p(x)$ exist; interior continuity is not required.

Show $p(1)=p(5)$ and $\lim_{x\to 3}p(x)=0$.