Limits of an increasing function Let $a&lt;b$ and let $f: [a,b] \to R$ be an increasing function.
I'm asked to provide an example where $\lim_{x\to a^{+}}f(x)$ does not equal
$$\inf\{\,f(x) \; | \;x \in [a,b]\,\},$$
and I'm having trouble understanding what exactly is going on in this situation and was wondering if anyone can enlighten me.
Background
Prior to this question I was asked to prove that there exists an $S$ in $R$ such that
$$ S = \inf\{\,f(x)\; |\; x \in (a,b]\,\}.$$
I used the fact that $F$ is an increasing function and showed $F$ is bounded from below and therefore must have an infimum.
Then I was asked to prove that the $\lim_{x\to a^{+}}f(x)$ exists, and is equal to S.
I showed this by definition of limit, using the fact that if $F$ is bounded from below, then there exists $L$ such that $L$ is less than or equal to $F(x)$.
I have a feeling my professor wants me to use the things I showed above to find the example to the initial question, but I just can't connect the dots…
This is a picture of what can happen at the endpoint (what needs to happen, in fact).

Remember to keep your counterexamples as simple as possible!
For the limit part ($L$ is the unfilled circle), you can perhaps use the sequential definition of limit, with the version of the completeness axiom that applies to monotonic sequences.
The big difference between the two infimums is that one includes $a$ while the other doesn't. To take advantage of it we use cases. For example,
$$
f(x) = \begin{cases}g(x), &amp;x \neq a\\
-1 &amp;x=  a\end{cases}
$$
What will happen if $g(x) &gt; -1\ \forall x \in (a,b]$?