Simple discontinuity of real valued function Consider a real valued function $f$ defined on $[a,b]$. Say it has a simple discontinuity at a point $x \in (a,b)$, with $f(x-) \neq f(x+)$ (LHL not equal to RHL). 

Is it necessary that $f(x)$ is equal to either $f(x-)$ or $f(x+)$, or can it be different from both?

Kindly help!
Consider the function 
$f(x) = 0$ when $x &lt; 2$;

$f(x) = 1$ when $x = 2$;

$f(x) = 2$ when  $x &gt; 2$;

Here, $f (2-) = 0;$
          $f (2+) = 2;$
          $f (2) = 1$
Consider the function 
$f(x) = 0$ when $x &lt; 2$;

$f(x) = 1$ when $x = 2$;

$f(x) = 2$ when  $x &gt; 2$;

Here, $f (2-) = 0;$
          $f (2+) = 2;$
          $f (2) = 1$