2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

In this guide, we will be discussing the connection between differentiability and continuity in the AP Calculus AB curriculum. 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.

Differentiability refers to the existence of a derivative for a given function at a point. A function is said to be differentiable at a point if its derivative exists at that point. If a function is differentiable at every point in its domain, it is considered to be a differentiable function. Continuity refers to the behavior of a function at a 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 continuous at every point in its domain, it is considered to be a continuous function.

A function that is differentiable at a point is also 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. Therefore, if the function is differentiable at a point, it must also be continuous at that point. However, a function that is continuous at a point does not necessarily have to be differentiable at that point. A function can be continuous at a point but not differentiable if its derivative is not defined or does not exist at that point. This is known as a point of discontinuity. For 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. Determining When Derivatives Do and Do Not Exist: There are several methods to determine when derivatives exist and do not exist for a given function. 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. Another method is to use the graph of the function. 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. 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. Example Problems: Example 1: Determine whether the function f(x) = x^2 is differentiable at the point x = 0. Solution: We can use the limit definition of the derivative to determine whether the function f(x) = x^2 is differentiable at x = 0. The limit of (f(x)-f(0))/(x-0) as x approaches 0 is (x^2 - 0^2)/x = x. 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. Example 2: Determine whether the function g(x) = |x| is differentiable at the point x = 0. Solution: We can use the limit definition of the derivative to determine whether the function g(x) = |x| is differentiable at x = 0. The limit of (g(x)-g(0))/(x-0) as x approaches 0 is (|x| - 0)/x = |x|/x. 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. Therefore, the derivative of g(x) = |x| does not exist at x = 0 and the function is not differentiable at that point.

Another way to determine if the function is differentiable at x=0 is by analyzing the graph of the function, we can see that the function has a sharp corner at x=0 and it does not have a smooth transition, therefore it is not differentiable at x=0.

It is also important to note that even though the function g(x) = |x| is not differentiable at x=0, it is still continuous at that point. This is because the limit of the function as x approaches 0 is equal to the value of the function at x=0 which is 0.

In conclusion, understanding the connection between differentiability and continuity is an important concept in Calculus as it allows us to determine the smoothness of a function and its behavior at certain points. To determine when derivatives exist and do not exist, we can use methods such as the limit definition of the derivative or analyzing the graph of the function.