Is this true that a continuous function in a right-closed interval is bounded on that interval?

Lets say that I have a function $f$ that is continuous on $[a_1,b_1]$ for some $a_1<b_1$. I know that I can conclude that $f$ is bounded of $[a_1,b_1]$ .

Is this also valid for right-closed intervals?

Let $g$ be a function that is continuous on $(a_2,b_2]$ for some $a_2<b_2$. Does this mean that $g$ is bounded on $(a_2,b_2]$ ?

I was able to prove that if $a$ is removable discontinuity then it is bounded on that interval.

On all other cases, lets say when $a$ is essential/jump discontinuity, does it mean that $g$ is at least bounded from above or below in that interval?

No, it is not true. Take, for instance$$\begin{array}{ccc}(0,1]&\longrightarrow&\Bbb R\\x&\mapsto&\displaystyle\frac1x\sin\left(\frac1x\right)\end{array}$$which is unbounded (from above and also from below).