Discontinuity of Logarithmic Functions

The logarithmic function is an important type of function in calculus and can exhibit discontinuities in certain cases.

Understanding the discontinuity of logarithmic functions is essential for mastering the concept of discontinuity.

Basic Properties

The logarithmic function has the following basic properties:

• Monotonicity: the logarithmic function is a monotone function
• Continuity: continuous on its domain
• Discontinuity: discontinuous at certain boundary points
• Inverse relation: the logarithmic function and the exponential function are inverses of each other

Basic Logarithmic Functions

The Natural Logarithmic Function

Properties:

• Domain: $(0, +\infty)$
• Continuous on its domain
• The graph is a monotonically increasing logarithmic curve
• Range: $\mathbb{R}$
• Discontinuous at $x = 0$ (the function is undefined there)

The General Logarithmic Function

Properties:

• Domain: $(0, +\infty)$
• Continuous on its domain
• Monotonically increasing when $a > 1$, monotonically decreasing when $0 < a < 1$
• Range: $\mathbb{R}$
• Discontinuous at $x = 0$ (the function is undefined there)

Analysis of the Points of Discontinuity of Logarithmic Functions

Point of Discontinuity: $x = 0$

Analysis:

• The function is undefined at $x = 0$
• Right limit: $\lim_{x \to 0^+} \ln x = -\infty$
• This is a discontinuity of the second kind

Characteristics:

• The function values tend to negative infinity
• There is a vertical asymptote $x = 0$
• The limit does not exist (it tends to infinity)

Discontinuity of Composite Logarithmic Functions

Example 1: $f(x) = \ln(x^2 - 1)$

Analysis:

• The inner function $h(x) = x^2 - 1$ is continuous on $\mathbb{R}$
• The outer function $g(x) = \ln x$ is continuous on $(0, +\infty)$
• We need $x^2 - 1 > 0$, that is, $x < -1$ or $x > 1$
• Therefore the interval of continuity is $(-\infty, -1) \cup (1, +\infty)$
• It is discontinuous at $x = -1$ and $x = 1$

Analysis of the points of discontinuity:

• Points of discontinuity: $x = -1$ and $x = 1$
• At these points, the inner function satisfies $x^2 - 1 = 0$
• The logarithmic function is undefined at 0
• These are boundary discontinuities

Example 2: $f(x) = \ln(\cos x)$

Analysis:

• The inner function $h(x) = \cos x$ is continuous on $\mathbb{R}$
• The outer function $g(x) = \ln x$ is continuous on $(0, +\infty)$
• We need $\cos x > 0$
• Therefore the interval of continuity is all the values for which $\cos x > 0$ holds, i.e. all the $x$ values that satisfy it
• It is discontinuous at every point where $\cos x = 0$

Analysis of the points of discontinuity:

• Points of discontinuity: $x = \frac{\pi}{2} + k\pi$ ($k \in \mathbb{Z}$)
• At these points, $\cos x = 0$
• The logarithmic function is undefined at 0
• These are periodic points of discontinuity

Example 3: $f(x) = \ln(x^2 + 1)$

Analysis:

• The inner function $h(x) = x^2 + 1$ is continuous on $\mathbb{R}$
• The outer function $g(x) = \ln x$ is continuous on $(0, +\infty)$
• Since $x^2 + 1 > 0$ holds for every $x \in \mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous on $\mathbb{R}$

Conclusion: this function is continuous everywhere on $\mathbb{R}$ and has no point of discontinuity.

Types of Point of Discontinuity

The points of discontinuity of logarithmic functions mainly fall into the following types:

1. Discontinuities of the second kind: the limit tends to infinity
2. Undefined points: the function is undefined at that point
3. Boundary discontinuities: discontinuous at the boundary of the domain

Determining Continuity

1. Basic logarithmic functions: continuous on $(0, +\infty)$
2. Composite logarithmic functions: the range of the inner function must be checked to lie within $(0, +\infty)$
3. Piecewise logarithmic functions: the left limit, the right limit and the function value at the breakpoint must be checked

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Exercises

Exercise 1

Determine the continuity of $f(x) = \ln(x^2 + 1)$ on $\mathbb{R}$.

Exercise 2

Determine the intervals of continuity of $f(x) = \ln(x^2 - 1)$.

Exercise 3

Determine the intervals of continuity of $f(x) = \ln(\cos x)$.

Exercise 4

Determine the type of discontinuity of $f(x) = \ln x$ at $x = 0$.

Exercise 5

Determine the continuity of $f(x) = \ln|x|$ at $x = 0$.

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Summary

Symbols Used in This Article