Differentiability Of Functions F And G

Real Analysis Discussion

Introduction

In real analysis, the concept of differentiability is crucial in understanding the behavior of functions. A function is said to be differentiable at a point if the limit of the difference quotient exists at that point. In this discussion, we will explore the differentiability of functions f and g, specifically focusing on the scenario where exactly one of the functions is differentiable at zero.

Notations and Definitions

Before we dive into the discussion, let's establish some notations and definitions.

• A function f is said to be differentiable at a point a if the limit of the difference quotient exists at that point, i.e.,

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

  exists.

• The derivative of a function f at a point a is denoted as f'(a) and is defined as

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

Exactly One of the Functions is Differentiable at Zero

We are given that exactly one of the functions f and g is differentiable at zero. This means that either f is differentiable at zero and g is not, or g is differentiable at zero and f is not.

Case 1: f is Differentiable at Zero and g is Not

Let's assume that f is differentiable at zero and g is not. This implies that the limit of the difference quotient of f exists at zero, but the limit of the difference quotient of g does not exist at zero.

Example 1:

Consider the functions f(x) = |x| and g(x) = |x| + 1. Both functions are not differentiable at zero, but f is differentiable at zero and g is not.

import numpy as np
def f(x):
return np.abs(x)
def g(x):
return np.abs(x) + 1
h = 0.0001
print((f(0 + h) - f(0)) / h)

print((g(0 + h) - g(0)) / h)

In this example, the difference quotient of f at zero exists, but the difference quotient of g at zero does not exist.

Case 2: g is Differentiable at Zero and f is Not

Now, let's assume that g is differentiable at zero and f is not. This implies that the limit of the difference quotient of g exists at zero, but the limit of the difference quotient of f does not exist at zero.

Example 2:

Consider the functions f(x) = |x| and g(x) = |x| + 1. Both functions are not differentiable at zero, but g is differentiable at zero and f is not.

import numpy as np
def f(x):
return np.abs(x)
def g(x):
return np.abs(x) + 1

h = 0.0001
print((g(0 + h) - g(0)) / h)

printf(0 + h) - f(0)) / h)

In this example, the difference quotient of g at zero exists, but the difference quotient of f at zero does not exist.

Case 3: Neither f nor g is Differentiable at Zero

Now, let's assume that neither f nor g is differentiable at zero. This implies that the limit of the difference quotient of both functions does not exist at zero.

Example 3:

Consider the functions f(x) = |x| and g(x) = |x| + 1. Both functions are not differentiable at zero.

import numpy as np
def f(x):
return np.abs(x)
def g(x):
return np.abs(x) + 1

h = 0.0001
print((f(0 + h) - f(0)) / h)

print((g(0 + h) - g(0)) / h)

In this example, the difference quotient of both functions at zero does not exist.

Conclusion

In conclusion, we have explored the differentiability of functions f and g, specifically focusing on the scenario where exactly one of the functions is differentiable at zero. We have provided examples for each of the three cases: f is differentiable at zero and g is not, g is differentiable at zero and f is not, and neither f nor g is differentiable at zero. These examples demonstrate the differentiability of functions and provide insight into the behavior of functions at a point.

References

• [1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• [2] Bartle, R. G. (2011). The Elements of Real Analysis. John Wiley & Sons.

Further Reading

• [1] Differentiability of Functions. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Differentiability
• [2] Real Analysis. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Real_analysis

Code

import numpy as np
def f(x):
return np.abs(x)
def g(x):
return np.abs(x) + 1

h = 0.0001
print((f(0 + h) - f(0)) / h)

print((g(0 + h) - g(0)) / h)
**Differentiability of Functions f and g: Q&A**
=====================================

**Real Analysis Discussion**
---------------------------

### Q&A

In this section, we will address some frequently asked questions related to the differentiability of functions f and g.

#### 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 exists at that point. A function is said to be differentiable at a point if the limit of the difference quotient exists at that point.

#### Q: Can a function be differentiable at a point but not continuous at that point?

A: Yes, it is possible for a function to be differentiable at a point but not continuous at that point. For example, the function f(x) = |x| is differentiable at x = 0 but not continuous at x = 0.

#### Q: Can a function be continuous at a point but not differentiable at that point?

A: Yes, it is possible for a function to be continuous at a point but not differentiable at that point. For example, the function f(x) = |x| is continuous at x = 0 but not differentiable at x = 0.

#### Q: What is the relationship between differentiability and the derivative?

A: The derivative of a function at a point is the limit of the difference quotient at that point. If a function is differentiable at a point, then the derivative exists at that point.

#### Q: Can a function have a derivative at a point but not be differentiable at that point?

A: No, if a function has a derivative at a point, then it is differentiable at that point.

#### Q: Can a function be differentiable at a point but have a derivative that is not continuous at that point?

A: Yes, it is possible for a function to be differentiable at a point but have a derivative that is not continuous at that point. For example, the function f(x) = x^2 sin(1/x) is differentiable at x = 0 but has a derivative that is not continuous at x = 0.

#### Q: Can a function have a derivative that is continuous at a point but not be differentiable at that point?

A: No, if a function has a derivative that is continuous at a point, then it is differentiable at that point.

### Conclusion

In conclusion, we have addressed some frequently asked questions related to the differentiability of functions f and g. We have discussed the relationship between differentiability and continuity, the relationship between differentiability and the derivative, and the relationship between differentiability and the continuity of the derivative.

### References

*   [1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
*   [2] Bartle, R. G. (2011). The Elements of Real Analysis. John Wiley & Sons.

### Further Reading

*   [1] Differentiability of Functions. (n.d.). Retrieved from <https://en.wikipedia.org/wiki/Differentiability>
*   [2] Real Analysis. (n.d.). Retrieved from <https://en.wikipedia.org/wiki/Real_analysis>

### Code

```python
import numpy as np

def f(x):
    return np.abs(x)

def g(x):
    return np(x) + 1

# Calculate the difference quotient of f at zero
h = 0.0001
print((f(0 + h) - f(0)) / h)

# Calculate the difference quotient of g at zero
print((g(0 + h) - g(0)) / h)
</code></pre>
<h3>Additional Resources</h3>
<ul>
<li>[1] Differentiability of Functions. (n.d.). Retrieved from <a href="https://en.wikipedia.org/wiki/Differentiability">https://en.wikipedia.org/wiki/Differentiability</a></li>
<li>[2] Real Analysis. (n.d.). Retrieved from <a href="https://en.wikipedia.org/wiki/Real_analysis">https://en.wikipedia.org/wiki/Real_analysis</a></li>
<li>[3] Calculus. (n.d.). Retrieved from <a href="https://en.wikipedia.org/wiki/Calculus">https://en.wikipedia.org/wiki/Calculus</a></li>
</ul>