To determine whether the function f(x,y)f(x, y)f(x,y) is continuous at (0,0)(0, 0)(0,0) and whether it is continuous everywhere, let’s analyze the function carefully.<\/p>\n\n\n\n
Given that: f(x,y)={(cos\u2061y)sin\u2061xif x\u22600cos\u2061yif x=0f(x, y) = \\begin{cases} (\\cos y) \\sin x & \\text{if } x \\neq 0 \\\\ \\cos y & \\text{if } x = 0 \\end{cases}f(x,y)={(cosy)sinxcosy\u200bif x\ue020=0if x=0\u200b<\/p>\n\n\n\n
1. Continuity at (0,0)(0, 0)(0,0):<\/h3>\n\n\n\n
To check continuity at (0,0)(0, 0)(0,0), we need to verify if the following condition holds: lim\u2061(x,y)\u2192(0,0)f(x,y)=f(0,0).\\lim_{(x, y) \\to (0, 0)} f(x, y) = f(0, 0).(x,y)\u2192(0,0)lim\u200bf(x,y)=f(0,0).<\/p>\n\n\n\n
First, calculate f(0,0)f(0, 0)f(0,0):
When x=0x = 0x=0, f(0,0)=cos\u2061(0)=1f(0, 0) = \\cos(0) = 1f(0,0)=cos(0)=1.<\/p>\n\n\n\n
Now, evaluate the limit lim\u2061(x,y)\u2192(0,0)f(x,y)\\lim_{(x, y) \\to (0, 0)} f(x, y)lim(x,y)\u2192(0,0)\u200bf(x,y) for different paths:<\/p>\n\n\n\n
- \n
- Path 1 (along x=0x = 0x=0):<\/strong>
Along this path, the function is f(0,y)=cos\u2061yf(0, y) = \\cos yf(0,y)=cosy. As y\u21920y \\to 0y\u21920, cos\u2061y\u21921\\cos y \\to 1cosy\u21921. Hence, lim\u2061(0,y)\u2192(0,0)f(0,y)=1\\lim_{(0, y) \\to (0, 0)} f(0, y) = 1lim(0,y)\u2192(0,0)\u200bf(0,y)=1.<\/li>\n\n\n\n- Path 2 (along y=0y = 0y=0):<\/strong>
Along this path, the function is f(x,0)=sin\u2061xf(x, 0) = \\sin xf(x,0)=sinx. As x\u21920x \\to 0x\u21920, sin\u2061x\u21920\\sin x \\to 0sinx\u21920. Hence, lim\u2061(x,0)\u2192(0,0)f(x,0)=0\\lim_{(x, 0) \\to (0, 0)} f(x, 0) = 0lim(x,0)\u2192(0,0)\u200bf(x,0)=0.<\/li>\n<\/ul>\n\n\n\nSince the limit depends on the path taken (0 along y=0y = 0y=0 and 1 along x=0x = 0x=0), the limit does not exist at (0,0)(0, 0)(0,0). Therefore, the function is not continuous at<\/strong> (0,0)(0, 0)(0,0).<\/p>\n\n\n\n
2. Continuity Everywhere:<\/h3>\n\n\n\n
Now, we need to check if f(x,y)f(x, y)f(x,y) is continuous everywhere.<\/p>\n\n\n\n
- \n
- When x\u22600x \\neq 0x\ue020=0, f(x,y)=(cos\u2061y)sin\u2061xf(x, y) = (\\cos y) \\sin xf(x,y)=(cosy)sinx, and both cos\u2061y\\cos ycosy and sin\u2061x\\sin xsinx are continuous functions. Thus, f(x,y)f(x, y)f(x,y) is continuous when x\u22600x \\neq 0x\ue020=0.<\/li>\n\n\n\n
- However, as we saw in the previous part, f(x,y)f(x, y)f(x,y) is not continuous at<\/strong> (0,0)(0, 0)(0,0), meaning that the function is not continuous at all points where x=0x = 0x=0.<\/li>\n<\/ul>\n\n\n\n
Thus, the function is not continuous everywhere<\/strong>.<\/p>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n
- \n
- f(x,y)f(x, y)f(x,y) is not continuous at (0,0)(0, 0)(0,0).<\/li>\n\n\n\n
- f(x,y)f(x, y)f(x,y) is not continuous everywhere.<\/li>\n<\/ul>\n\n\n\n
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<\/figure>\n","protected":false},"excerpt":{"rendered":"(cos y) sin x If x \u00e2\u2030 0, let f(x, y) = cos y If x = 0, is f continuous at (0, 0)? Is f continuous everywhere? Justify your answers. The Correct Answer and Explanation is: To determine whether the function f(x,y)f(x, y)f(x,y) is continuous at (0,0)(0, 0)(0,0) and whether it is continuous everywhere, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-257648","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257648","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=257648"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/257648\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=257648"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=257648"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=257648"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Path 2 (along y=0y = 0y=0):<\/strong>