# Intermediate Value Theorem Example and Statement

Here, you will learn intermediate value theorem example and statement and single point continuity.

## Intermediate Value Theorem Example with Statement

Statement : Suppose f(x) is continuous on an interval I, and a and b are any two points of I. Then if $$y_0$$ is a number between f(a) and f(b), there exist a number c between a and b such that f(c) = $$y_0$$.

Note that a function f which is continuous in [a,b] possesses the following properties :

(i) If f(a) & f(b) posses opposite signs, then there exists at least one root of the equation f(x) = 0 in the open interval (a,b).

(ii) If K is any real number between f(a) & f(b), then there exist atleast one root of the equation f(x) = K in the open interval (a,b).

Example : Show that the function, f(x) = $$(x – a)^2$$ $$(x – b)^2$$ + x, takes the value $$a + b\over 2$$ for some $$x_0$$ $$\in$$ (a, b)

Solution : f(x) = $$(x – a)^2$$ $$(x – b)^2$$ + x

f(a) = a   &   f(b) = b

& $$(a + b)\over 2$$ $$\in$$ (f(a), f(b))

$$\therefore$$   By intermediate value theorem, there is at least one $$x_0$$ $$\in$$ (a, b) such that f($$x_0$$) = $$(a + b)\over 2$$

## Some Important Points on Continuity

(a) If f(x) is continuous & g(x) is discontinuous at x = a then the product function $$\phi (x)$$ = f(x).g(x) will not necessarily be discontinuous at x = a,

For e.g.  f(x) = x  &  g(x) = $$sin{\pi\over x}$$ at x $$\ne$$ 0 and g(x) = 0 at  x = 0

f(x) is continuous at x = 0 & g(x) is discontinuous at x = 0, but f(x).g(x) is continuous at x = 0.

(b) If f(x) and g(x) both are discontinuous at x = a then the product function $$\phi (x)$$ = f(x).g(x) is not necessarily be discontinuous at x = a,

For e.g.  f(x) = -g(x) = [1 at x $$\geq$$ 0  -1 at x < 0]

f(x) & g(x) both are discontinuous at x = 0, but the product function f(x).g(x) is still continuous at x = 0, but the product function f(x).g(x) is still continuous at x = 0.

(c) If f(x) and g(x) both are discontinuous at x = a then f(x) $$\pm$$ g(x) is not necessarily be discontinuous at x = a.

(d) A continuous function whose domain is closed must have a range also in closed interval.

(e) If f is continuous at x = a & g is continuous at x = f(a) then the composite g[f(x)] is continuous at x = a.

## Single Point Continuity

Functions which are continuous only at one point are said to exhibit single point continuity.