![]() This limit does not exist as x approaches 0 because as x approaches 0 from the left the limit is -1 and as x approaches 0 from the right the limit is 1. We can use the limit definition of the derivative to determine whether the function g(x) = |x| is differentiable at x = 0. ![]() This limit is finite and non-infinite, therefore the derivative of f(x) = x^2 exists at x = 0 and the function is differentiable at that point.ĭetermine whether the function g(x) = |x| is differentiable at the point x = 0. We can use the limit definition of the derivative to determine whether the function f(x) = x^2 is differentiable at x = 0. However, if the function has a "corner" or "break" in its graph, its derivative will not exist at the points where the "corner" or "break" occurs.ĭetermine whether the function f(x) = x^2 is differentiable at the point x = 0. If the function is smooth and has no "corners" or "breaks" in its graph, then its derivative will exist at all points in its domain. One method is to use the limit definition of the derivative, which states that the derivative of a function f(x) at a point x=a exists if the limit of (f(x)-f(a))/(x-a) as x approaches a is finite and non-infinite.Īnother method is to use the graph of the function. There are several methods to determine when derivatives exist and do not exist for a given function. This is known as a point of discontinuity.įor example, a function that has a sharp "corner" or a "break" in its graph, such as a step function or piecewise function, will be continuous but not differentiable at the points where the corner or break occurs.ĭetermining When Derivatives Do and Do Not Exist: A function can be continuous at a point but not differentiable if its derivative is not defined or does not exist at that point. However, a function that is continuous at a point does not necessarily have to be differentiable at that point. Therefore, if the function is differentiable at a point, it must also be continuous at that point. This is because if the derivative of a function exists at a point, the limit of the function as x approaches that point is equal to the value of the derivative at that point. ![]() If a function is continuous at every point in its domain, it is considered to be a continuous function.Ĭonnecting Differentiability and Continuity:Ī function that is differentiable at a point is also continuous at that point. A function is said to be continuous at a point if the limit of the function as x approaches that point is equal to the value of the function at that point. ![]() If a function is differentiable at every point in its domain, it is considered to be a differentiable function.Ĭontinuity refers to the behavior of a function at a point. A function is said to be differentiable at a point if its derivative exists at that point. Understanding when derivatives exist and do not exist is an important concept in Calculus as it allows us to determine the smoothness of a function and its behavior at certain points.ĭefining Differentiability and Continuity:ĭifferentiability refers to the existence of a derivative for a given function at a point. Condition 2 fails at \(x=3\).In this guide, we will be discussing the connection between differentiability and continuity in the AP Calculus AB curriculum.
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