Q.Discuss the continuity of $f$ on its domain, where $f(x) = |x+1|$, for $-3 \le x \le 2$, $= |x-5|$, for $2 < x \le 7$.

$f(x)=|x+1|$ for $-3\le x\le2$, and $f(x)=|x-5|$ for $2<x\le7$. Each piece, being the modulus of a continuous linear function, is continuous on its own sub-interval; the only point to check is the junction $x=2$.

$f(2)=|2+1|=3$ (first piece, which includes $x=2$).

Left-hand limit: $\displaystyle\lim_{x\to2^-}|x+1|=|2+1|=3$ (trivial, since the first piece is continuous).

Right-hand limit: $\displaystyle\lim_{x\to2^+}|x-5|=|2-5|=|-3|=3$. …

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