Is $g$ increasing? 
Suppose $f:[a,b]\rightarrow\mathbb{R}$ and define $g:[a,b]\rightarrow\mathbb{R}$ as follows
  $$
g(x)=\sup\{f(t):a\leq t\leq x\},
$$
  with $x\in[a,b]$.

Someone told me that $g$ is an increasing function, but I don't see it. I don't see, how in the case of $g$, $\forall y_1,y_2\in[a,b]$, $$y_1\leq y_2 \rightarrow g(y_1)\leq g(y_2).$$
Please help me out with this. Thanks!

It is increasing (not necessarily strictly) because you are taking supremum over an increasingly bigger set.
