Is an unbounded function bounded on a bounded non-compact interval?

Question

I'm a little confused about functions in the set of bounded continuous functions.

For example, if we take the interval (0,1] and the function

$f(x) = $\begin{cases} 0 & x \in (0,\frac{1}{2}]\\ t + \frac{1}{2} & x \in (\frac{1}{2},1]\\ \end{cases}

Then is this function in the set of bounded continuous functions?

Alternatively, if $f(x)=x^\frac{-1}{2}$ on I=(0,1] then is this function in the set of bounded continuous functions?

Thanks for the help in advance.