Step 1:
At $x=1$
LHL=$\lim\limits_{\large x\to 1^-}f(x)=\lim\limits_{\large x\to 1^-}(x+5)$
$\qquad\qquad\qquad\;=(1+5)$
$\qquad\qquad\qquad\;=6$
RHL=$\lim\limits_{\large x\to 1^+}f(x)=\lim\limits_{\large x\to 1^+}(x-5)$
$\qquad\qquad\qquad\;=(1-5)$
$\qquad\qquad\qquad\;=(-4)$
$f(1)=1+5$
$\;\;\;\;\;\;\;=6$
f(1)=LHL $\neq$ RHL
$f$ is not continuous at $x=1$
Step 2:
At $x=c < 1$
$\lim\limits_{\large x\to c}(x+5)=c+5$
$\qquad\quad\quad\;\;=f(c)$
At $x=c > 1$
$\lim\limits_{\large x\to c}(x-5)=c-5$
$\qquad\quad\quad\;\;=f(c)$
Step 3:
$f$ is continuous at all points $x\in R$ except $x=1$.