The function f(x) = 1 + |sin x| is

$f(x) = 1 + |\sin x|$

so for $\sin x ≥ 0 ⇒ f (x) = 1 + \sin x$

$\sin x ≤ 0 ⇒ f(x) = 1 − \sin x$

for $\underset{x→0^+}{\lim}f(x)=1$

$\underset{x→0^-}{\lim}f(x)=1$

so f(x) is continuous for all $x∈R$