Understanding Continuity and Differentiability in Functions

Not every function is continuously differentiable, and it is important to distinguish between the concepts of differentiability and uniformly continuous functions. This article clarifies these distinctions, explores counterexamples, and concludes with a discussion on the Weierstrass function as a classic illustrative case.

Continuous Differentiability vs. Differentiability

A function f(x) is called differentiable at a point a if the derivative f'(a) exists. This means that the limit defining the derivative must exist. However, a function is said to be continuously differentiable on an interval if it is differentiable at every point in that interval and its derivative is a continuous function on that interval.

Counterexamples for Continuously Differentiable Functions

Absolute Value Function: The function f(x) |x| is differentiable everywhere except at x 0 where the derivative does not exist. Therefore, f(x) |x| is not continuously differentiable.

Piecewise Functions: A piecewise function can also be differentiable at certain points but may not have a continuous derivative. Consider the function:

a(x) begin{cases}text{ }x^2 text{if } x

This function is differentiable everywhere, but the derivative a'(x) is not continuous at x 1.

Uniform Continuity and Differentiability

It is also crucial to note that not all uniformly continuous functions are differentiable. Uniform continuity and differentiability are, in fact, distinct properties of functions.

Basic Uniform Continuity Example: Consider the function f: mathbb{R} rightarrow mathbb{R} defined by f(x) x. This function is uniformly continuous on mathbb{R} because given any epsilon > 0, let delta epsilon. Then for all x, y in mathbb{R} where x - y , the reverse triangle inequality yields:

[|f(x) - f(y)| |x - y| leq |x - y|

However, f(x) x is not differentiable on mathbb{R}; it is well-known that f(x) |x| (the absolute value function) is not differentiable at x 0 as shown in the following proof:

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

However, this limit does not exist because the one-sided limits are not equal:

[lim_{x to 0^{pm}} frac{x}{x} lim_{x to 0^{pm}} frac{pm x}{x} pm 1]

Nowhere Differentiable Functions

A notable example of a uniformly continuous function that is nowhere differentiable is the Weierstrass function, a classic illustrative case in real analysis. The Weierstrass function is constructed using the sum of cosine functions with increasing frequencies and decreasing amplitudes. This function is remarkably smooth (it is continuous everywhere) but has no derivative at any point.

Conclusion

To sum up, while many functions, such as polynomials and sine/cosine functions, are continuously differentiable, there are numerous examples of functions that are differentiable but not continuously differentiable. Additionally, functions can be uniformly continuous without being differentiable, exemplified by the Weierstrass function. These concepts are crucial in understanding the nuanced properties of functions in mathematical analysis.