Differential calculus
Properties of continuous functions
Properties of continuous functions
Continuous function definition
A real function  y = f (x)  is continuous at a point a if it is defined at x = a and
that is, if for every e > 0 there is a  d(e) > 0 such that  | f (x) - f (a) | < e  whenever  | x - a | < d(e).
Therefore, if a function changes gradually as independent variable changes, so that at every value a, of the independent variable, the difference between f (x) and f (a) approaches zero as x approaches a.
Thus, a function is continuous at a point if both one-sided limits (the left-handed and right-handed limits) are
 equal,
that is, if it is continuous both on the left and on the right at that point.
A point at which the function value is not equal to its limit, as x approaches that point, is called point of discontinuity. A function having points of discontinuity is discontinuous.
A function is said to be continuous if it is continuous at all points
1. Intermediate value theorem - Bolzano's theorem
If a function is continuous on the closed interval [a, b] then it takes every value between  f (a) and  f (b) for at least one argument between a and b.
That is, for every  y between  f (a) and  f (b) there exists at least one argument x between a and b whose function's value  f (x) = y, as shows the figure below.
2. Existence of roots
If a function is continuous on the closed interval [a, b] then, if  f (a) and  f (b) have opposite signs, or f (a) ·  f (b) < 0, then there exists x Î [a, b] such that  f (x) = 0.
That is, the function has at least one real root. For example, inside the closed interval [a, b], the function shown in the figure below, has three roots,  x1, x2 and x3,
that is,    f (x1) = 0,  f (x2) = 0 and  f (x3) = 0  since  f(a) ·  f(b) < 0.
3. Extreme value theorem
If a function is continuous on the closed interval [a, b] then, there exist  xmin and  xMax Î [a, b] such that
for all  x Î [a, b],  values of the function   f (xmin) <  f (x) <  f (xMax).
4. Monotone function
A continuous function is monotone on an interval if it is consistently increasing or decreasing in value,
so that either                    f (x1) <  f (x2  for all   x1 < x2,
or                     f (x1) >  f (x2  for all   x1 < x2
These may be called strictly monotone functions to distinguish them from those satisfying either
f (x1) <  f (x2  for all   x1 < x2,
or                     f (x1) >  f (x2  for all   x1 < x2,
that are called weakly monotone.
Calculus contents C