Continuity and Differentiability

This calculus video tutorial provides a basic introduction into continuity and differentiability. Continuity tells you if the function f(x) is continuous or discontinuous at some point in the interval (a,b). Points of discontinuity include holes, jump discontinuities, and infinite discontinuities. A hole is considered a removable discontinuity. Jump and infinite discontinuities are considered nonremovable discontinuities. You can determine if a function is continuous using the 3-step continuity test. Jump discontinuities occur when there's a disconnection in the graph and infinite discontinuities occur at vertical asymptotes. Differentiability describes the continuity of the first derivative function f'(x). If a function is differentiable at a point, then it is continuous at that point. If a function is continuous at a point, it may or may not be differentiable at that point. If a function is discontinuous at a point, then it is not differentiable at that point. To determine the differentiability of a function, you can use the 3-step continuity test on f'(x) using limits. You need to make sure that the left and right one sided limits are equal to each other. If they're not, the function is not continuous and there is a jump discontinuity or an infinite discontinuity the one sided limits equal positive and negative infinity. If the one sided limits are equal to each other, you need to make sure that the function f(x) is defined at that it equals the limit as x approaches c of f(x), otherwise there will be a hole in the graph. This video contains plenty of examples and practice problems of analyzing the continuity and differentiability of f(x) using piecewise functions as an example. This video also contains the graphs of piecewise functions so you can visually see if the function is continuous and/or differentiable. Differentiable functions have a smooth curve. If there are any sharp turns, cusps, or vertical tangents, the function will not be differentiable at those points.

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Видео Continuity and Differentiability канала The Organic Chemistry Tutor