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) = \((x โ a)^2\) \((x โ b)^2\) + x, takes the value \(a + b\over 2\) for some \(x_0\) \(\in\) (a, b)
Solution : f(x) = \((x โ a)^2\) \((x โ b)^2\) + x
f(a) = a ย & ย f(b) = b
& \((a + b)\over 2\) \(\in\) (f(a), f(b))
\(\therefore\) ย 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.ย f(x) = xย &ย g(x) = \(sin{\pi\over x}\) at x \(\ne\) 0 and g(x) = 0 atย 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.ย f(x) = -g(x) = [1 at x \(\geq\) 0ย -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.
