Variable Interval

Recall that variable $$${x}$$$ is defined by the set $$${X}$$$ of those values that it can take (in considered question). This set $$${X}$$$ that contains each value of $$${x}$$$ exactly once is called domain of variation of the variable $$${x}$$$.

In general any numerical set can be domain of variability.

Recall that geometrically we treat numbers as points on numerical axis. Domain $$${X}$$$ of variation of variable $$${x}$$$ on this axis can be represented as some set of points. Therefore, numerical values of $$${x}$$$ are also called points.

Often we deal with variable, whose domain of variation is set $$$\mathbb{N}$$$ of all natural numbers.

For sequence $$${x}_{{n}}=\frac{{{1}+{{\left(-{1}\right)}}^{{n}}}}{{n}}$$$ domain of the variation is set of fractions of the form $$$\frac{{1}}{{m}}$$$ (when $$${m}={1},{2},{3},\ldots$$$) and number 0.

For constant quantity domain of variation consists of one number.

However, in calculus we study variables, that change continuously, like distance, time etc. Domain of variation of such variable is numerical interval. Often it is finite interval that is bounded by numbers $$${a}$$$ and $$${b}$$$ $$$\left(a< b\right)$$$.

There are three such types of intervals:

  • Closed interval $$${\left[{a},{b}\right]}:\ {a}\le{x}\le{b}$$$.
  • Half-opened intervals $$${\left({a},{b}\right]}:\ {a}<{x}\le{b}$$$ and $$${\left[{a},{b}\right)}:\ {a}\le{x}<{b}$$$.
  • Open interval $$${\left({a},{b}\right)}:{a}<{x}<{b}$$$.

In all three cases length of interval is $$${b}-{a}$$$.

Geometrically interval is represented by segment of numerical axis.

Also we need to consider infinite intervals, which have one or both represented by special numbers $$$-\infty$$$ and $$$+\infty$$$.

For example, $$${\left(-\infty,\infty\right)}$$$ is set of real numbers; $$${\left({a}+\infty\right)}$$$ is set of values $$${x}$$$ that satisfy inequality $$${x}>{a}$$$; interval $$${\left(-\infty,{b}\right]}$$$ is set of values $$${x}$$$ that satisfy inequality $$${x}\le{b}$$$.

Geometrically infinite intervals are represented by open ray.