{"id":3844,"date":"2021-08-09T10:06:54","date_gmt":"2021-08-09T10:06:54","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3844"},"modified":"2021-11-21T01:06:15","modified_gmt":"2021-11-20T19:36:15","slug":"intermediate-value-theorem","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/intermediate-value-theorem\/","title":{"rendered":"Intermediate Value Theorem Example and Statement"},"content":{"rendered":"

Here, you will learn intermediate value theorem example and statement and single point continuity.<\/p>\n

Intermediate Value Theorem Example with Statement<\/h2>\n

Statement :<\/strong> 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\\).<\/p>\n

Note that a function f which is continuous in [a,b] possesses the following properties :<\/p>\n

\n

(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).<\/p>\n

(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).<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Show that the function, f(x) = \\((x – a)^2\\) \\((x – b)^2\\) + x, takes the value \n\t \\(a + b\\over 2\\) for some \\(x_0\\) \\(\\in\\) (a, b)<\/p>\n\t

Solution : <\/span>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\\)

<\/p>\n\n\n

Some Important Points on Continuity<\/h2>\n

(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,<\/p>\n

For e.g.  f(x) = x  &  g(x) = \\(sin{\\pi\\over x}\\) at x \\(\\ne\\) 0 and g(x) = 0 at  x = 0<\/p>\n

f(x) is continuous at x = 0 & g(x) is discontinuous at x = 0, but f(x).g(x) is continuous at x = 0.<\/p>\n

(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,<\/p>\n

For e.g.  f(x) = -g(x) = [1 at x \\(\\geq\\) 0  -1 at x < 0]<\/p>\n

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.<\/p>\n

(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.<\/p>\n

(d) A continuous function whose domain is closed must have a range also in closed interval.<\/p>\n

(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.<\/p>\n

Single Point Continuity<\/h2>\n

Functions which are continuous only at one point are said to exhibit single point continuity.<\/p>\n\n\n

\n
Previous – Types of Discontinuities \u2013 Removable and Nonremovable<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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 …<\/p>\n

Intermediate Value Theorem Example and Statement<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[20],"tags":[191,193,192],"yoast_head":"\nIntermediate Value Theorem Example and Statement - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn intermediate value theorem example and statement and some results on continuity.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/intermediate-value-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Intermediate Value Theorem Example and Statement - 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