Consider a function $f : (a, b) \rightarrow X$, where $X$ is a metric space. We say that the right-hand limit of $f$ at $x \in [a, b)$ is $q$ if for every $x_n \rightarrow x$ in $(x, b)$, $f(x_n) \rightarrow q$.
$$ f(x^+) = q \iff \text{For all sequences $\{x_n\}$ in $(x, b)$, $x_n \rightarrow x \implies f(x_n) \rightarrow q$} $$
Moreover, a function is called right-continuous if for all $x \in (a,b)$, $f(x^+) = f(x)$. The notions of a left limit and left-continuity are defined analogously.
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Limit Exists iff One-Sided Limits Agree.
$$ \lim_{x \rightarrow p} f(x) \text{ exists} \iff f(p^+) = f(p^-) = \lim _{x \rightarrow p} f(x) $$
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Consider a function $f : (a, b) \rightarrow X$ where $X$ is a metric space. We say that $f$ is discontinuous at $x$ if $f$ is not continuous at $x$. Moreover, we distinguish between two types of discontinuities:
$$ \textbf{Of the 1st Kind } \space \text{ $f(x^+) \neq f(x^-)$ or $f(x^+) = f(x^-) \neq f(x)$ but both limits exist.} \\
\textbf{Of the 2nd Kind } \space \text{Either $f(x^+)$ or $f(x^-)$ does not exist.} \hspace{100pt} $$
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At Most Countable 1st Kind Discontinuities. The set of points for which $f : (a, b) \rightarrow \R$ has a discontinuity of the 1st kind is at most countable.
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A function $f : (a, b) \rightarrow \R$ is said to be monotone increasing on $(a, b)$ if the following holds:
$$ a < x < y<b \implies f(x) \leq f(y) $$
Note that because of this property, monotonic functions only have discontinuities of the 1st kind.
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Increasing if Negation Decreasing. A function $f$ is monotone increasing iff $-f$ is monotone decreasing.
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One-Sided Limits. Let $f:(a,b)\rightarrow \R$ be monotone increasing on $(a,b)$. Then we have the following,
$$ \sup _{a < t <x}\{f(t)\} = f(x^-) \leq f(x) \leq f(x^+) = \inf _{x < t< b}\{f(t)\} $$
Moreover, monotonicity is preserved in these limits.
$$ a<x\leq y<b \implies f(x^+) \leq f(y^-) $$
An analogous result holds if $f$ is monotone decreasing on $(a,b)$.