In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one...
In mathematics, a subset
A
{\displaystyle A}
of a topological space is said to be dense-in-itself if
A
{\displaystyle A}
contains no isolated points.
Every dense-in-itself closed set is perfect. Conversely, every perfect set is dense-in-itself.
A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number
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First we will define the left and right derivatives as follows.
Left derivative of $f(x)$: $f(x)^+ = \lim_{h \to 0^+} \frac {f(x+h) - \lim_{a \to x^+} f(a)}{h}$
Right derivative of $f(x)$: $f(x)^- = \lim_{h \to 0^-} \frac {f(x+h) - \lim_{a \to x^-} f(a)}{h}$
Now let us define the set of piecew...
