Uniformly Continuous Function sending Bounded Set to Unbounded One
Solution 1:
Consider the space $X\subset \ell^\infty$ consisting of elements of the form $(1,\ldots,1,t,0,\ldots)$ such that $t\in [0,1]$. Note that $X$ is bounded under the $\sup$ metric. Define the function $f:X\to \mathbb R$ by $f((x_n))=\sum\limits_{n=1}^\infty x_n$. The function $f$ is uniformly continuous, as given any $\epsilon>0$ if we let $\delta=\max\{\epsilon,1\}$ we have that $\|x-y\|<\delta\implies |f(x)-f(y)|<\epsilon$ for $x,y\in X$. Yet $f(X)$ is clearly unbounded.