{"id":3840,"date":"2021-08-09T09:30:20","date_gmt":"2021-08-09T09:30:20","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3840"},"modified":"2021-11-21T01:07:41","modified_gmt":"2021-11-20T19:37:41","slug":"continuity-of-a-function","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/continuity-of-a-function\/","title":{"rendered":"Definition of Continuity of a Function"},"content":{"rendered":"

This tutorial is for you if you are searching for – “Definition of Continuity of a function, Discontinuity and Missing Point Discontinuity.”<\/p>\n

Let’s begin –<\/p>\n

Continuous Functions<\/h2>\n

A Function for which a small change in the independent variable causes only a small change and not a sudden jump in the dependent variable are called continuous functions. Naively, we may say that a function is continuous at a fixed point if we can draw the graph of the function around that point without lifting the pen from the plane of the paper.<\/p>\n

Continuity of a function at a point<\/h2>\n

A function f(x) is said to be continuous at x = a, if<\/p>\n

\n

\\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = f(a). <\/strong><\/p>\n<\/blockquote>\n

Symbolically f is continuous at x = a if<\/p>\n

\n

\\(\\displaystyle{\\lim_{h \\to 0}}\\) f(a – h) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) f(a + h) = f(a), h > 0 <\/strong><\/p>\n

i.e. \\(LHL_{x = a}\\) = \\(RHL_{x = a}\\) equals value of ‘f’ at x = a.<\/strong><\/p>\n<\/blockquote>\n\n\n

Example : <\/span> If f(x) = {\\(sin{\\pi x\\over 2}\\), x < 1 and [x], x \\(\\geq\\) 1} then find whether f(x) is continuous or not at x = 1, where [ ] denotes greatest integer function.<\/p>\n\t

Solution : <\/span>For continuity at x = 1, we determine, f(1), \\(\\displaystyle{\\lim_{x \\to {1^-}}}\\) f(x) and \\(\\displaystyle{\\lim_{x \\to {1^+}}}\\) f(x)

Now, f(1) = [1] = 1

    \\(\\displaystyle{\\lim_{x \\to {1^-}}}\\) f(x) = \\(\\displaystyle{\\lim_{x \\to {1^-}}}\\) \\(sin{\\pi x\\over 2}\\) = \\(sin{\\pi\\over 2}\\) = 1 and \\(\\displaystyle{\\lim_{x \\to {1^+}}}\\) f(x) = \\(\\displaystyle{\\lim_{x \\to {1^+}}}\\) [x] = 1

so   f(1) = \\(\\displaystyle{\\lim_{x \\to {1^-}}}\\) f(x) = \\(\\displaystyle{\\lim_{x \\to {1^+}}}\\) f(x)

\\(\\therefore\\)   f(x) is continuous at x = 1.

<\/p>\n\n\n

Continuity of a function in an interval<\/h2>\n

(a) A function is said to be continuous in (a,b) if f is continuous at each & every point belonging to (a, b).<\/p>\n

(b) A function is said to be continuous in a closed interval [a,b] if :<\/p>\n

\n

(i) f is continuous in the open interval (a,b)<\/p>\n

(ii) f is right continuous at ‘a’ i.e. \\(\\displaystyle{\\lim_{x \\to {a^+}}}\\) f(x) = f(a) = a finite quantity.<\/p>\n

(iii) f is left continuous at ‘b’ i.e. \\(\\displaystyle{\\lim_{x \\to {b^-}}}\\) f(x) = f(b) = a finite quantity.<\/p>\n<\/blockquote>\n

Note :<\/strong><\/p>\n

(i) All polynomials, trigonometrical functions, exponential & logarithmic functions are continuous in their domains.<\/p>\n

(ii) If f(x) & g(x) are two functions that are continuous at x = c then the function defined by:<\/p>\n

\\(F_1(x)\\) = f(x) + g(x); \\(F_2(x)\\) = Kf(x), where K is any real number; \\(F_3(x)\\) = f(x).g(x) are also continuous at x = c.<\/p>\n

Further, if g(c) is not zero, then \\(F_4(x)\\) = \\(f(x)\\over g(x)\\) is also continuous at x = c.<\/p>\n

Hope, you learnt definition of continuity of a function and continuity of a function at a point and over an interval. Practice more question on continuity of a function to learn more and get ahead in competition. Good Luck!<\/p>\n\n\n

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

This tutorial is for you if you are searching for – “Definition of Continuity of a function, Discontinuity and Missing Point Discontinuity.” Let’s begin – Continuous Functions A Function for which a small change in the independent variable causes only a small change and not a sudden jump in the dependent variable are called continuous …<\/p>\n

Definition of Continuity of a Function<\/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":[196,194,195,197],"yoast_head":"\nDefinition of Continuity of a Function - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn definition of continuity of a function and continuity of a function at a point and over an interval.\" \/>\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\/continuity-of-a-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Definition of Continuity of a Function - 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