The Relationship Between Differentiability and Continuity

Oct 13, 2025 by windailys.com 23 views

Introduction

When studying the topic of differentiability of multivariable functions, one of the most fundamental concepts to grasp is the relationship between differentiability and continuity. In this article, we will delve into the world of differentiable functions in $\mathbb{R}^n$ and explore where we can "see" continuity.

What is Differentiability?

A function $f: \mathbb{R}^n \to \mathbb{R}$ is said to be differentiable at a point $p \in \mathbb{R}^n$ if there exists a linear transformation $T: \mathbb{R}^n \to \mathbb{R}$ such that

\lim_{h \to 0} \frac{|f(p + h) - f(p) - T(h)|}{||h||} = 0

where $h \in \mathbb{R}^n$ and $||h||$ denotes the Euclidean norm of $h$ . The linear transformation $T$ is called the derivative of $f$ at $p$ and is denoted by $Df(p)$ .

The Relationship Between Differentiability and Continuity

One of the most important results in multivariable calculus is that differentiability implies continuity. In fact, we have the following theorem:

Theorem 1: If a function $f: \mathbb{R}^n \to \mathbb{R}$ is differentiable at a point $p \in \mathbb{R}^n$ , then $f$ is continuous at $p$ .

Proof: Let $f$ be differentiable at $p$ . Then, by definition, there exists a linear transformation $T: \mathbb{R}^n \to \mathbb{R}$ such that

\lim_{h \to 0} \frac{|f(p + h) - f(p) - T(h)|}{||h||} = 0

Now, let $\epsilon > 0$ be given. Choose $\delta = \epsilon / (||T|| + 1)$ . Then, for any $h \in \mathbb{R}^n$ with $||h|| < \delta$ , we have

|f(p + h) - f(p)| \leq |f(p + h) - f(p) - T(h)| + |T(h)| \leq \epsilon ||h|| + ||T|| ||h|| < \epsilon ||h|| + \epsilon ||h|| = 2 \epsilon ||h||

Since $||h|| < \delta$ , we have $2 \epsilon ||h|| < 2 \epsilon \delta = \epsilon$ . Therefore, $|f(p + h) - f(p)| < \epsilon$ for all $h \in \mathbb{R}^n$ with $||h|| < \delta$ . This shows that $f$ is continuous at $p$ .

The Relationship Between Continuity and Differentiability

The converse of Theorem 1 is not necessarily true. In other words, a function can be continuous at a point without being differentiable at that point. However, we do have the following result:

Theorem 2: If a function $f: \mathbb{R}^n \to \mathbb{R}$ is continuously differentiable at a point $p \in \mathbb{R}^n$ , then $f$ is differentiable at $p$ .

Proof: Let $f$ be continuously differentiable at $p$ . Then, by definition, the derivative $Df(p)$ exists and is continuous at $p$ . Now, let $h \in \mathbb{R}^n$ be given. We need to show that

\lim_{h \to 0} \frac{|f(p + h) - f(p) - Df(p)(h)|}{||h||} = 0

Since $Df(p)$ is continuous at $p$ , we have

\lim_{h \to 0} \frac{|Df(p)(h)|}{||h||} = 0

Therefore, for any $\epsilon > 0$ , there exists $\delta > 0$ such that

\frac{|Df(p)(h)|}{||h||} < \epsilon

whenever $||h|| < \delta$ . Now, let $h \in \mathbb{R}^n$ be given with $||h|| < \delta$ . We have

|f(p + h) - f(p) - Df(p)(h)| = |f(p + h) - f(p) - Df(p)(h) + Df(p)(h) - Df(p)(h)|

\leq |f(p + h) - f(p) - Df(p)(h)| + |Df(p)(h) - Df(p)(h)|

\leq |f(p + h) - f(p) - Df(p)(h)| + \epsilon ||h||

\leq \epsilon ||h|| + \epsilon ||h|| = 2 \epsilon ||h||

Therefore, we have

\frac{|f(p + h) - f(p) - Df(p)(h)|}{||h||} \leq \frac{2 \epsilon ||h||}{||h||} = 2 \epsilon

Since $\epsilon$ was arbitrary, we have

\lim_{h \to 0} \frac{|f(p + h) - f(p) - Df(p)(h)|}{||h||} = 0

This shows that $f$ is differentiable at $p$ .

Conclusion

In conclusion, we have seen that differentiability implies continuity, but the converse is not necessarily true. However, if a function is continuously differentiable at a point, then it is differentiable at that point. This result highlights the importance of continuity in the study of differentiability.

References

[1] Spivak, M. (1965). Calculus on manifolds. W.A. Benjamin.
[2] Rudin, W. (1976). Principles of mathematical analysis. McGraw-Hill.
[3] Lang, S. (1983). Real and functional analysis. Springer-Verlag.

Introduction

In our previous article, we explored the relationship between differentiability and continuity in the context of multivariable functions. We saw that differentiability implies continuity, but the converse is not necessarily true. However, if a function is continuously differentiable at a point, then it is differentiable at that point. In this article, we will answer some frequently asked questions about differentiability and continuity.

Q: What is the difference between differentiability and continuity?

A: Differentiability and continuity are related but distinct concepts. A function is continuous at a point if the limit of the function as the input approaches that point exists and is equal to the value of the function at that point. A function is differentiable at a point if the limit of the difference quotient as the input approaches that point exists and is equal to the derivative of the function at that point.

Q: Can a function be continuous at a point without being differentiable at that point?

A: Yes, a function can be continuous at a point without being differentiable at that point. For example, the function $f(x) = |x|$ is continuous at $x = 0$ but not differentiable at $x = 0$ .

Q: Can a function be differentiable at a point without being continuously differentiable at that point?

A: Yes, a function can be differentiable at a point without being continuously differentiable at that point. For example, the function $f(x) = x^2 \sin(1/x)$ is differentiable at $x = 0$ but not continuously differentiable at $x = 0$ .

Q: What is the relationship between differentiability and the existence of a derivative?

A: A function is differentiable at a point if and only if the derivative of the function at that point exists. In other words, if a function is differentiable at a point, then the derivative of the function at that point exists, and if the derivative of the function at a point exists, then the function is differentiable at that point.

Q: Can a function have multiple derivatives at a point?

A: Yes, a function can have multiple derivatives at a point. For example, the function $f(x) = x^2$ has a second derivative at $x = 0$ .

Q: What is the relationship between differentiability and the existence of a Taylor series?

A: A function is differentiable at a point if and only if the Taylor series of the function at that point converges. In other words, if a function is differentiable at a point, then the Taylor series of the function at that point converges, and if the Taylor series of a function at a point converges, then the function is differentiable at that point.

Q: Can a function be differentiable at a point without being analytic at that point?

A: Yes, a function can be differentiable at a point without being analytic at that point. For example, the function $f(x) = e^x$ is differentiable at $x = 0$ but not analytic at $x = 0$ .

Conclusion

In conclusion, we have seen that differentiability and continuity are related but distinct concepts. A function can be continuous at a point without being differentiable at that point, but if a function is continuously differentiable at a point, then it is differentiable at that point. We have also seen that a function can have multiple derivatives at a point and that the existence of a derivative is equivalent to differentiability.

References

[1] Spivak, M. (1965). Calculus on manifolds. W.A. Benjamin.
[2] Rudin, W. (1976). Principles of mathematical analysis. McGraw-Hill.
[3] Lang, S. (1983). Real and functional analysis. Springer-Verlag.