Differentiability Of Functions F And G

Introduction

In real analysis, the concept of differentiability plays a crucial role in understanding the behavior of functions. A function is said to be differentiable at a point if its derivative exists at that point. In this article, we will explore the differentiability of functions f and g, and examine the conditions under which they are differentiable or not differentiable at a point, specifically at zero.

The Problem

We are given two functions, f and g, defined on all of R. We need to determine the conditions under which these functions are not differentiable at zero. Specifically, we need to find examples of functions f and g that are not differentiable at zero, but where the product fg is differentiable at zero.

The Conditions for Differentiability

A function f is said to be differentiable at a point a if the following limit exists:

$$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$

This limit is denoted as f'(a) and is called the derivative of f at a.

For a function to be differentiable at a point, it must satisfy the following conditions:

1. Continuity: The function must be continuous at the point.
2. Existence of the Limit: The limit must exist.

If a function satisfies these conditions, then it is said to be differentiable at that point.

Examples of Functions Not Differentiable at Zero

Let's consider some examples of functions that are not differentiable at zero.

Example 1: Absolute Value Function

Consider the absolute value function f(x) = |x|. This function is not differentiable at zero because the limit does not exist.

$$\lim_{h \to 0} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0} \frac{|h|}{h}$$

This limit does not exist because the left-hand limit is -1 and the right-hand limit is 1.

Example 2: Signum Function

Consider the signum function f(x) = sgn(x), which is defined as:

$$\begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}$$

This function is not differentiable at zero because the limit does not exist.

Example 3: Piecewise Function

Consider the piecewise function f(x) = {x^2 if x < 0, x if x >= 0}. This function is not differentiable at zero because the left-hand limit is 0 and the right-hand limit is 1.

Examples of Functions Differentiable at Zero

Now, let's consider some examples of functions that are differentiable at zero.

Example 1: Linear Function

Consider the linear function f(x) = x. This function is differentiable at zero because the limit exists.

$$\lim_{h \to 0} \frac{(0 + h) - 0}{h} = \lim_{h \to 0} \frac{h}{h} = 1$$

Example 2: Quadratic Function

Consider the quadratic function f(x) = x^2. This function is differentiable at zero because the limit exists.

$$\lim_{h \to 0} \frac{(0 + h)^2 - 0^2}{h} = \lim_{h \to 0} \frac{h^2}{h} = 0$$

Example 3: Exponential Function

Consider the exponential function f(x) = e^x. This function is differentiable at zero because the limit exists.

$$\lim_{h \to 0} \frac{e^{0 + h} - e^0}{h} = \lim_{h \to 0} \frac{e^h - 1}{h} = 1$$

Conclusion

In conclusion, we have seen that the differentiability of functions f and g at zero depends on the conditions under which they are defined. We have also seen that there are examples of functions that are not differentiable at zero, but where the product fg is differentiable at zero. These examples demonstrate the importance of considering the product of functions when analyzing their differentiability.

References

• [1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• [2] Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley.
• [3] Spivak, M. (1965). Calculus. W.A. Benjamin.

Further Reading

For further reading on the topic of differentiability, we recommend the following resources:

• [1] Khan Academy: Differentiability
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram MathWorld: Differentiability
  Differentiability of Functions f and g: Q&A =====================================================

Introduction

In our previous article, we explored the differentiability of functions f and g, and examined the conditions under which they are differentiable or not differentiable at a point, specifically at zero. In this article, we will answer some frequently asked questions related to the differentiability of functions f and g.

Q&A

Q: What is the difference between differentiability and continuity?

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

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

A: No, a function cannot be differentiable at a point without being continuous at that point. Differentiability implies continuity, but continuity does not necessarily imply differentiability.

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

A: The derivative of a function is a measure of how fast the function changes as its input changes. If a function is differentiable at a point, then its derivative exists at that point. Conversely, if the derivative of a function exists at a point, then the function is differentiable at that point.

Q: Can a function be differentiable at a point if its derivative is not defined at that point?

A: No, a function cannot be differentiable at a point if its derivative is not defined at that point. The derivative of a function is a measure of how fast the function changes as its input changes, and if the derivative is not defined at a point, then the function is not differentiable at that point.

Q: What is the significance of the differentiability of a function at a point?

A: The differentiability of a function at a point is significant because it implies that the function can be approximated by a linear function near that point. This is useful in many applications, such as optimization and machine learning.

Q: Can a function be differentiable at a point if it has a discontinuity at that point?

A: No, a function cannot be differentiable at a point if it has a discontinuity at that point. A discontinuity at a point implies that the function is not continuous at that point, and therefore it is not differentiable at that point.

Q: What is the relationship between the differentiability of a function and its domain?

A: The differentiability of a function is related to its domain. A function can only be differentiable at points in its domain where it is continuous. If a function has a discontinuity at a point, then it is not differentiable at that point.

Q: Can a function be differentiable at a point if it has a vertical tangent at that point?

A: No, a function cannot be differentiable at a point if it has a vertical tangent at that point. A vertical tangent at a point implies that the function is not continuous at that point, and therefore it is not differentiable at that point.

Q: What is the significance of the differentiability of a function at a point in optimization?

A: The differentiability of a function at a point is significant in optimization because it implies that the function can be approximated by a linear function near that point. This is useful in many optimization algorithms, such as gradient descent.

Q: Can a function be differentiable at a point if it has a local maximum or minimum at that point?

A: Yes, a function can be differentiable at a point if it has a local maximum or minimum at that point. In fact, the differentiability of a function at a point is a necessary condition for it to have a local maximum or minimum at that point.

Q: What is the relationship between the differentiability of a function and its second derivative?

A: The differentiability of a function is related to its second derivative. If a function is differentiable at a point, then its second derivative exists at that point. Conversely, if the second derivative of a function exists at a point, then the function is differentiable at that point.

Conclusion

In conclusion, we have answered some frequently asked questions related to the differentiability of functions f and g. We hope that this article has provided a useful resource for those interested in the differentiability of functions.

References

• [1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• [2] Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley.
• [3] Spivak, M. (1965). Calculus. W.A. Benjamin.

Further Reading

For further reading on the topic of differentiability, we recommend the following resources:

• [1] Khan Academy: Differentiability
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram MathWorld: Differentiability