Introduction

The relationship between analyticity, continuity, and differentiability is a fundamental topic in calculus and real analysis. Understanding these concepts and their interrelations is crucial for both theoretical and practical applications in mathematics and beyond. This article aims to clarify the implications of analyticity and differentiability on the continuity of a function.

Understanding Differentiability Implies Continuity

Differentiability is a stronger property than continuity. If a function is differentiable at a point, it must be continuous at that point. This relationship can be explained using the definition of the derivative and the properties of limits.

Mathematical Proof of Differentiability Implies Continuity

Consider a function (f) that is differentiable at (x_0). By definition, this means that the limit of the difference quotient exists:

[lim_{h to 0} frac{f(x_0 h) - f(x_0)}{h} L text{ for some } L in mathbb{R}]

Let's denote this difference quotient by (delta(h)). Therefore:

[delta(h) frac{f(x_0 h) - f(x_0)}{h}]

We can write:

[f(x_0 h) f(x_0) delta(h) h]

Since (lim_{h to 0} delta(h) L), we can say:

[lim_{h to 0} f(x_0 h) f(x_0) L cdot 0 f(x_0)]

This shows that (f) is continuous at (x_0). The continuity of (f) at (x_0) is guaranteed by the differentiability of (f) at (x_0).

Ultralimit Approach to Demonstrating Differentiability Implies Continuity

An alternative proof of differentiability implying continuity can be given using the ultralimit concept. By taking an ultrapower of the real numbers with respect to a nonprincipal ultrafilter, one can work with additional infinitesimals, which provides another perspective on continuity and differentiability.

I have written a detailed PDF writeup on this topic, which can be found here.

Remarks and Examples

While differentiability implies continuity, the converse is not true. A classic example is the function (f(x) x) which is continuous everywhere but not differentiable at (x 0), because the limit defining the derivative does not exist at zero. Another famous example is the Weierstrass function, which is continuous everywhere but differentiable nowhere.

These examples highlight that continuity is a necessary but not sufficient condition for differentiability. Other issues, such as vertical tangents or sharp corners, can prevent a function from being differentiable even if it is continuous.

The Implications of Analyticity

Analytic functions are infinitely differentiable and can be expressed as power series in a neighborhood of any point within their domain. Analyticity is a stronger property than differentiability. All analytic functions are continuous, but not all continuous functions are analytic.

Counterexamples to the Converse

To further illustrate, consider the function (f(x) |x|). This function is continuous at (x 0) but not differentiable there due to the sharp corner at (x 0).

The Weierstrass function is a more extreme example, demonstrating that continuity does not guarantee differentiability. It is continuous everywhere but differentiable nowhere, showcasing the disconnect between continuity and differentiability.

Conclusion

In summary, the relationship between analyticity, continuity, and differentiability is best understood via their definitions and examples. While differentiability implies continuity, the converse does not hold. Care must be taken when interpreting these concepts to avoid confusion.

For a deeper exploration of these concepts, please refer to the detailed research and examples cited throughout this article.

References

Ultrapower of Real Numbers: Ultrafilter and Non-standard Analysis