Understanding a Function Continuous at Exactly One Point and Not Differentiable There: A Unique Example
When discussing the behavior of functions, continuity and differentiability are two fundamental concepts. A function is continuous at a point if the limit of the function as x approaches that point exists and is equal to the function's value at that point. Conversely, a function is differentiable at a point if it has a derivative at that point, indicating the presence of a well-defined tangent with a finite slope.
Discontinuous at One Point: The Absolute Value Function
One classic example of a function that is continuous but not differentiable at a specific point is the absolute value function. The absolute value function is defined as:
f(x) |x|
Contiuity: This function is continuous for all real numbers because its graph has no breaks or jumps. For all x ≠ 0, the one-sided limits as x approaches 0 from the left and right are the same as the function's value there. However, at x 0, the one-sided limits from the left and right are not identical, causing a discontinuity in the derivative.
Proof of Non-Differentiability at x 0
To demonstrate this non-differentiability, consider the definition of the derivative:
f'(x) limh→0 [f(x h) - f(x)] / h
For the absolute value function at x 0, the left-hand and right-hand limits of the difference quotient are different:
Left-hand limit as h → 0-: limh→0- [(0 h) - 0] / h limh→0- h / h 1
Right-hand limit as h → 0 : limh→0 [(0 h) - 0] / h limh→0 h / h 1
Because the left and right limits are not equal, the derivative does not exist at x 0. This contradiction demonstrates that the absolute value function is continuous but not differentiable at x 0.
Other Examples of Continuous but Not Differentiable Functions
Functions that are continuous but not differentiable can have sharp corners, cusps, or vertical tangents with infinite slopes. A sawtooth function is an example of such a function that is differentiable from both the left and the right at all points. However, an infinite number of semicircles can create a function that is continuous everywhere but differentiable nowhere, such as the Weierstrass function.
For such a function, it is established that:
f(x) x
This function, f(x) x, is continuous everywhere but not differentiable at x 0. As you can see:
If x 0, f’(0) -1 and if x 0, f’(0) 1. For the function to be differentiable at x 0, both one-sided limits would need to converge to the same value.
Conclusion
Despite the simplicity of the example, understanding the nuances between continuity and differentiability is crucial in advanced calculus. Functions like the absolute value and the sawtooth function exemplify the diverse ways in which these properties can manifest in mathematical functions. For a deeper dive, you may want to explore related questions and more intricate functions like the Weierstrass function.