I recognize intuitively what it means for a function to be continuous (i.e. no jumps or breaks in the function), but the concept of being uniformly continuous seems to be over my head.
I'm looking at the following function:
$f(x)=\frac{2^x}{x}$, $D=(0,5]$
How can I determine if this function is uniformly continuous or not?
A fast, intuitive, graphical approach to detect uniform continuity is to use the fact that a uniformly continuous real-valued function on a subset $A$ of the reals can be uniquely extended to a continuous function on the closure of $A$. In your case, the function blows up as $x$ tends to 0, so there is no way to continuously extend the function to $[0,5]$. Hence the function is not uniformly continuous.
All of this can be formalized using the definition of uniformly continuity.