https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:continuity-and-differentiability-ncert-new/x6f66cbee9a2f805b:continuity-of-different-functions/v/continuity-of-polynomial-functions 2025-05-22T13:12:03.767544849Z Continuity of polynomial functions In this video, we'll take things from concrete to abstract. We'll start by testing continuity at two points for a simple polynomial functions. We'll then test the continuity of that function across its domain. We'll finally zoom out and test the continuity of all polynomial functions for all points in their domain. Before this final step, we'll take a refresher on what polynomial functions are along with looking at a number of examples. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Continuity of polynomial functions In this video, we'll take things from concrete to abstract. We'll start by testing continuity at two points for a simple polynomial functions. We'll then test the continuity of that function across its domain. We'll finally zoom out and test the continuity of all polynomial functions for all points in their domain. Before this final step, we'll take a refresher on what polynomial functions are along with looking at a number of examples. https://cdn.kastatic.org/ka-youtube-converted/gvpIZuFz9uk.mp4/gvpIZuFz9uk.mp4 452 Continuity of different functions https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:continuity-and-differentiability-ncert-new/x6f66cbee9a2f805b:continuity-of-different-functions/v/continuity-of-rational-functions 2025-05-22T13:12:03.767544849Z Continuity of rational functions In this video, we'll discuss continuity of two rational functions. Before we solve problems, we'll refresh the definition of rational functions. For the first function, we find the limits at x = 0 and x = 5 for 1 by (x-5). For the second function, we find the limits at x = -5 for (x^2 - 25) by (x + 5). In both of these problems, we're looking at points where the function is not defined. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Continuity of rational functions In this video, we'll discuss continuity of two rational functions. Before we solve problems, we'll refresh the definition of rational functions. For the first function, we find the limits at x = 0 and x = 5 for 1 by (x-5). For the second function, we find the limits at x = -5 for (x^2 - 25) by (x + 5). In both of these problems, we're looking at points where the function is not defined. https://cdn.kastatic.org/ka-youtube-converted/1BzT-BtxpKc.mp4/1BzT-BtxpKc.mp4 475 Continuity of different functions https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:continuity-and-differentiability-ncert-new/x6f66cbee9a2f805b:continuity-of-different-functions/v/continuity-of-piecewise-defined-functions 2025-05-22T13:12:03.767544849Z Continuity of piecewise defined functions In this video, we'll discuss continuity of piecewise defined functions. We'll first look at a problem that involves polynomial functions. We then cover two more problems involinv mod and constant functions. For each of these, we'll test the continuity of the function where its definition changes. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Continuity of piecewise defined functions In this video, we'll discuss continuity of piecewise defined functions. We'll first look at a problem that involves polynomial functions. We then cover two more problems involinv mod and constant functions. For each of these, we'll test the continuity of the function where its definition changes. https://cdn.kastatic.org/ka-youtube-converted/omTbdXFuxZo.mp4/omTbdXFuxZo.mp4 501 Continuity of different functions https://www.khanacademy.org/math/revision-term-1-ncert-math-class-12/x6a634f0b79812b7f:week-2/x6a634f0b79812b7f:continuity-and-differentiability/v/making-a-piecewise-defined-function-continuous 2025-05-22T13:12:03.767544849Z Making a piecewise defined function continuous In this video, we'll solve a problems where we already know that the given function is continuous. Using its continuity, we'll find values of a few variables. For the first problem, we'll find the relationship between two variables. For the next one, we'll find the values of two variables. For the last one, we'll encounter scenarios which are always true and never true. Its interesting to see how we report the solution in such scenarios. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Making a piecewise defined function continuous In this video, we'll solve a problems where we already know that the given function is continuous. Using its continuity, we'll find values of a few variables. For the first problem, we'll find the relationship between two variables. For the next one, we'll find the values of two variables. For the last one, we'll encounter scenarios which are always true and never true. Its interesting to see how we report the solution in such scenarios. https://cdn.kastatic.org/ka-youtube-converted/MKmLeZsDJQU.mp4/MKmLeZsDJQU.mp4 550 Continuity of different functions https://www.khanacademy.org/math/revision-term-1-ncert-math-class-12/x6a634f0b79812b7f:week-2/x6a634f0b79812b7f:continuity-and-differentiability/e/making-a-piecewise-defined-function-continuous 2025-08-08T02:45:51.78671331Z Making a piecewise defined function continuous Making a piecewise defined function continuous Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:continuity-and-differentiability-ncert-new/x6f66cbee9a2f805b:continuity-of-different-functions/v/continuity-of-greatest-integer-function 2025-05-22T13:12:03.767544849Z Continuity of greatest integer function In this video, we'll talk about continuity of greatest integer function. We'll first plot its graph and try to figure out points of discontinuity visually. We'll then make separate cases - for integers and non-integers - and evaluate continuity for both. Finally, we'll solve a problem where we use the definition of GIF to prove discontinuity at integral points of a different function. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Continuity of greatest integer function In this video, we'll talk about continuity of greatest integer function. We'll first plot its graph and try to figure out points of discontinuity visually. We'll then make separate cases - for integers and non-integers - and evaluate continuity for both. Finally, we'll solve a problem where we use the definition of GIF to prove discontinuity at integral points of a different function. https://cdn.kastatic.org/ka-youtube-converted/xMBSE2OlvyA.mp4/xMBSE2OlvyA.mp4 521 Continuity of different functions https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:continuity-and-differentiability-ncert-new/x6f66cbee9a2f805b:continuity-of-different-functions/v/algebra-of-continuous-functions 2025-05-22T13:12:03.767544849Z Algebra of continuous functions In this video, we'll cover algebra of continuous functions. Just like we could do things in the world of limits, we can add, subtract, multiply, and divide continuous functions. The resultant functions are also continuous wherever they are defined. We prove this for the case of addition before solving a worked example. We also take a look at how the algebra of continous functions works in the world of trigonometry. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Algebra of continuous functions In this video, we'll cover algebra of continuous functions. Just like we could do things in the world of limits, we can add, subtract, multiply, and divide continuous functions. The resultant functions are also continuous wherever they are defined. We prove this for the case of addition before solving a worked example. We also take a look at how the algebra of continous functions works in the world of trigonometry. https://cdn.kastatic.org/ka-youtube-converted/W5qy1D75gtU.mp4/W5qy1D75gtU.mp4 427 Continuity of different functions https://www.khanacademy.org/math/revision-term-1-ncert-math-class-12/x6a634f0b79812b7f:week-2/x6a634f0b79812b7f:continuity-and-differentiability/v/continuity-of-trigonometric-functions 2025-05-22T13:12:03.767544849Z Continuity of trigonometric functions We prove using limits the continuity of trigonometric functions. We use trigonometric identities to prove continuity for sin(x) and cos(x). We use algebra of limits to prove continuity for tan(x) and cot(x). We also figure out the points of discontinuity of functions other than sin(x) and cos(x). Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Continuity of trigonometric functions We prove using limits the continuity of trigonometric functions. We use trigonometric identities to prove continuity for sin(x) and cos(x). We use algebra of limits to prove continuity for tan(x) and cot(x). We also figure out the points of discontinuity of functions other than sin(x) and cos(x). https://cdn.kastatic.org/ka-youtube-converted/vcuaBcQZnkk.mp4/vcuaBcQZnkk.mp4 527 Continuity of different functions https://www.khanacademy.org/math/revision-term-1-ncert-math-class-12/x6a634f0b79812b7f:week-2/x6a634f0b79812b7f:continuity-and-differentiability/e/continuity-of-trigonometric-functions 2025-08-08T02:45:51.78671331Z Continuity of trigonometric functions Continuity of trigonometric functions Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:continuity-and-differentiability-ncert-new/x6f66cbee9a2f805b:continuity-of-different-functions/v/continuity-of-composite-functions 2025-05-22T13:12:03.767544849Z Continuity of composite functions In this video, we understand how to check continuity of composite functions. We do this using the example of sin(x^2). We first visually observe the continuity. We then find the continuity at a point, then generalise it for all composite functions. Finally, we use this technique to check continuity of functions involving modulus and trigonometric functions. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Continuity of composite functions In this video, we understand how to check continuity of composite functions. We do this using the example of sin(x^2). We first visually observe the continuity. We then find the continuity at a point, then generalise it for all composite functions. Finally, we use this technique to check continuity of functions involving modulus and trigonometric functions. https://cdn.kastatic.org/ka-youtube-converted/8m9CSg2N8jM.mp4/8m9CSg2N8jM.mp4 408 Continuity of different functions