The concept of differentiability is fundamental in calculus, and it's closely tied to the smoothness of a function's graph. A function is differentiable at a point if its derivative exists at that point, which essentially means the function has a well-defined tangent line at that point. However, the presence of discontinuities, particularly removable discontinuities, can raise questions about differentiability. This article delves into the relationship between removable discontinuities and differentiability, exploring why a function with a removable discontinuity is not differentiable at the point of discontinuity.
Removable Discontinuities and Differentiability
A removable discontinuity arises when a function has a hole at a specific point on its graph. This hole occurs because the function is undefined at that point, but the limit of the function as it approaches that point exists. In essence, the discontinuity can be "removed" by redefining the function to take on the value of the limit at the point of discontinuity.
Understanding Differentiability
Differentiability at a point implies that the function's graph has a well-defined tangent line at that point. The tangent line represents the instantaneous rate of change of the function at that point. This rate of change is captured by the derivative of the function. However, the existence of a tangent line, and therefore differentiability, hinges on the function's behavior in the immediate vicinity of the point in question.
The Impact of Removable Discontinuities
Let's consider a function f(x) that has a removable discontinuity at x = a. This means that the limit of f(x) as x approaches a exists, but f(a) is undefined. While we can "remove" the discontinuity by redefining the function to have the value of the limit at x = a, this redefinition does not automatically make the function differentiable at x = a.
Why Differentiability Fails at a Removable Discontinuity
The key reason why a function with a removable discontinuity is not differentiable at the point of discontinuity is that the function's behavior near the point of discontinuity does not satisfy the requirements for the existence of a derivative. Specifically:
- The function is undefined at the point of discontinuity: Since the function is undefined at x = a, we cannot calculate the derivative directly using the standard definition of a derivative.
- The function's behavior near the point of discontinuity is not continuous: Even though the limit exists, the function "jumps" at x = a. This "jump" prevents the existence of a unique tangent line at x = a.
Illustrative Example
Consider the function f(x) = (x^2 - 1)/(x - 1). This function has a removable discontinuity at x = 1. Notice that:
- The function is undefined at x = 1.
- The limit of f(x) as x approaches 1 is 2. This can be seen by simplifying the function: f(x) = (x + 1) for x ≠ 1.
To "remove" the discontinuity, we can redefine the function as:
g(x) = { (x^2 - 1)/(x - 1) if x ≠ 1 { 2 if x = 1
Even though g(x) is now defined at x = 1, it is still not differentiable at x = 1. This is because the function's behavior near x = 1 is still discontinuous. There is a "jump" in the function's graph at x = 1, preventing the existence of a tangent line at that point.
Conclusion
While a removable discontinuity can be "removed" by redefining the function, this redefinition does not automatically make the function differentiable at the point of discontinuity. The presence of a removable discontinuity signifies a "jump" in the function's graph, preventing the existence of a unique tangent line and therefore precluding differentiability at that point. In essence, the discontinuity reflects a lack of smoothness in the function's graph, a requirement for differentiability. Consequently, a function with a removable discontinuity remains non-differentiable at the point of discontinuity, even after the discontinuity is "removed."