Univariate Calculus: Continuity and Differentiability

Name: Univariate Calculus: Continuity and Differentiability
Uploaded: 2025-01-30T11:07:32.000Z
Duration: 48 min 10 s

Machine Learning Foundations: Recap of Univariate Calculus

Overview of Continuity and Differentiability

The lecture begins with a recap of continuity and differentiability, focusing on real-valued one-dimensional functions f: mathbbR to mathbbR .

A function f is continuous at a point x^* in mathbbR if for all sequences converging to x^* , the function values converge to f(x^*) .

This can be expressed compactly as:

If x_i to x^* , then f(x_i) to f(x^*).

Example 1: Continuity of f(x) = x^2

Testing continuity at x^* = 2:

Any sequence converging to 2 should have its function values converge to 4.

An example sequence converging to 2 is given (e.g., 3, 2.5, 2.25), showing that as these approach 2, their squares approach 4.

Concludes that the function f(x)=x^2 is continuous everywhere.

Example 2: Discontinuity in Piecewise Function

Introduces a piecewise function defined as:

f(x) = -1 if x < 0,

f(0)=0,

f(x)=1 if x > 0.

Two sequences are considered that both converge to zero but yield different limits for their respective function values:

First sequence yields constant value of one.

Second sequence yields constant value of minus one.

Analysis of Discontinuity

The first sequence converges to zero while the second does not match this limit, violating the definition of continuity.

Even if we redefine the value at zero (e.g., setting it to +1), discontinuity remains due to conflicting limits from different sequences.

General Definition and Examples

A function is continuous at all points in its domain; thus, it’s classified as continuous overall only when this holds true universally across its domain.

Provides an example where the function is continuous everywhere (f(x)=x^2) versus another example which demonstrates discontinuity (f(x)) due to undefined behavior at certain points.

Additional Examples

Discusses another piecewise-defined function:

Defined as:

For x > 1,;f(x)=2x+1

For x ≤ 1,;f(x)=3

Understanding Continuity and Differentiability in Functions

The Concept of Infinity and Function Limits

Infinity is not a valid real number; thus, defining limits around it can be problematic. For instance, the sequence x_i = 1/2, 1/4, ldots converges to zero while f(x_i) = 1, 2, 4, 8, ldots does not converge.

The function f(x) = 1/x is continuous only when restricted to positive values. However, including zero in its domain results in discontinuity.

Examples of Discontinuous Functions

Another example is f(x) = sin(1/x) . Using the same sequence as before shows that while x_i to 0 , f(x_i) = cos(1), cos(2), ... does not converge.

This demonstrates that both functions discussed are examples of discontinuous functions due to their behavior near zero.

Differentiability Defined

A function f: R to R is differentiable at a point x^* in R if the limit

[

lim_xto x^* (f(x)-f(x^))/(x-x^)

]

exists. If this limit exists, it represents the derivative denoted as f'(x^*).

If a function is not continuous at a point x^* , it cannot be differentiable there. However, continuity does not guarantee differentiability.

Non-Differentiable Continuous Functions

An example of a continuous but non-differentiable function is f(x)=|x| . At x^*=0, different sequences approaching zero yield different limits for the derivative expression.

Specifically:

For positive sequences: Limit converges to 1.

For negative sequences: Limit converges to -1.

This discrepancy indicates that the derivative does not exist at this point.

Further Examples of Differentiability

Consider two piecewise functions:

Function A:

If x ≥ 2: f(x)=4x+2

If x < 2: f(x)=2x+8

This function is discontinuous at x^*=2.

Function B:

If x ≥ 2: f(x)=4x+2

If x < 2: f(x)=2x+6

This function is continuous but still non-differentiable at that point due to differing left-hand and right-hand limits.

Understanding Derivatives Through Limits

Definition of the Derivative

The derivative f'(x^) is defined as the limit of the difference quotient as x approaches x^ .

This can be expressed mathematically as:

[

f'(x^) = lim_x to x^ f(x) - f(x^)/x - x^

]

An alternative expression for the derivative is:

[

f'(x^) = lim_h to 0 f(x^ + h) - f(x^*)/h

]

Interpretation of Limits

The second expression is often easier to interpret, especially when considering sequences approaching zero (e.g., h = 1/2, 1/4, ...).

For example, if we set x^* = a , then we can visualize points on a graph corresponding to these limits.

Visualizing Derivatives with Triangles

By plotting points such as (x^, f(x^)) , (x^* + h, f(x^* + h)), and others, one can form a triangle that helps in understanding the slope.

The vertical side length represents the change in function values: f(x^* + h) - f(x^*).

Reducing Interval Size

As we reduce h, for instance from 1 to 0.5 or smaller fractions, we observe how this affects our triangle's dimensions and thus our slope approximation.

Continuing this reduction leads us closer to finding the true derivative value.

Convergence Towards Derivative

As h approaches zero, the line connecting points becomes more accurate in representing the function's behavior at that point.