Understanding Continuous but Non-Differentiable Functions

The concept of a function being continuous but not differentiable at certain points is a fundamental idea in calculus. This article delves into the nuances of such functions, providing clear explanations and key examples to help practitioners, students, and enthusiasts better understand this intriguing mathematical phenomenon.

Introduction to Continuous but Non-Differentiable Functions

When a function is continuous at a specific point, it means that the function's value at that point does not abruptly change. However, being non-differentiable at a point suggests that the slope of the tangent line to the function at that point is undefined. This situation often occurs due to sharp turns, kinks, or other inconsistencies in the function's behavior.

Example: ( f(x) |x| )

Consider the absolute value function ( f(x) |x| ), defined as:

[ f(x) begin{cases} x text{if } x geq 0 -x text{if } x 0 end{cases} ]

At ( x 0 ), the function is continuous but not differentiable. To see why, consider the behavior of the function around ( x 0 ).

Left and Right Hand Limits

The left-hand limit as ( x ) approaches 0 is:

[ lim_{x to 0^-} frac{f(x) - f(0)}{x - 0} lim_{x to 0^-} frac{-x - 0}{x} -1 ]

The right-hand limit as ( x ) approaches 0 is:

[ lim_{x to 0^ } frac{f(x) - f(0)}{x - 0} lim_{x to 0^ } frac{x - 0}{x} 1 ]

These limits are not equal, indicating that the derivative of ( f(x) |x| ) at ( x 0 ) does not exist. This sharp transition at ( x 0 ) results in a kink, making it impossible to draw a consistent tangent line.

General Concept: Differentiability vs. Continuity

The terms "continuous" and "differentiable" have precise meanings in calculus:

Continuous at a point: The function's value at that point is not abrupt or discontinuous. Differentiable at a point: The function is smooth at that point, meaning the derivative exists.

A function that is continuous at a point but not differentiable at that point may exhibit a sharp turn or a kink. This notion is best illustrated by the absolute value function, as well as other functions such as the modulus function ( y |x| ).

Example: Weierstrass Function

The Weierstrass function is a famous example of a function that is continuous everywhere but differentiable nowhere. This function is highly irregular and was introduced by Karl Weierstrass in the late 19th century. It is defined as:

[ W(x) sum_{n0}^{infty} a^n cos(b^n pi x) ]

The Weierstrass function is continuous on the entire real line but its graph is so jagged that it has no tangent at any point. This makes ( W(x) ) a fascinating example of a function that challenges our intuition about smoothness in calculus.

Conclusion

Understanding continuous but non-differentiable functions is crucial in calculus and analysis. Functions like the absolute value and the Weierstrass function provide valuable insights into the complexities of mathematical functions and their behavior. By exploring these examples, we can gain a deeper appreciation for the subtleties of calculus and the rich variety of functions that exist in mathematical theory.