Discontinuity of Exponential Functions

The exponential function is an important type of function in calculus and can exhibit discontinuities in certain cases. Understanding the discontinuity of exponential functions is essential for mastering the concept of discontinuity.

Basic Properties

The exponential function has the following basic properties:

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

Basic Exponential Functions

The Natural Exponential Function

Properties:

• Domain: $\mathbb{R}$
• Continuous on $\mathbb{R}$
• The graph is a monotonically increasing exponential curve
• Range: $(0, +\infty)$

The General Exponential Function

Properties:

• Domain: $\mathbb{R}$
• Continuous on $\mathbb{R}$
• Monotonically increasing when $a > 1$, monotonically decreasing when $0 < a < 1$
• Range: $(0, +\infty)$

Discontinuity of Composite Exponential Functions

Example 1: $f(x) = e^{\frac{1}{x}}$

Analysis:

• The inner function $h(x) = \frac{1}{x}$ is continuous at $x \neq 0$
• The outer function $g(x) = e^x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous at $x \neq 0$
• It is discontinuous at $x = 0$ (the inner function is undefined at that point)

Analysis of the point of discontinuity:

• Point of discontinuity: $x = 0$
• The function is undefined at that point
• This is an undefined point

Example 2: $f(x) = a^{\ln x}$ ($a > 0, a \neq 1$)

Analysis:

• The inner function $h(x) = \ln x$ is continuous on $(0, +\infty)$
• The outer function $g(x) = a^x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous on $(0, +\infty)$
• It is discontinuous at $x = 0$ (the inner function is undefined at that point)

Analysis of the point of discontinuity:

• Point of discontinuity: $x = 0$
• The function is undefined at that point
• This is a boundary discontinuity

Example 3: $f(x) = e^{\sin x}$

Analysis:

• The inner function $h(x) = \sin x$ is continuous on $\mathbb{R}$
• The outer function $g(x) = e^x$ is continuous on $\mathbb{R}$
• 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.

Discontinuity of Piecewise Exponential Functions

Example: A Piecewise Function

Let $f(x) = \begin{cases} e^x, & x \leq 0 \\ x + 1, & x > 0 \end{cases}$

Analysis:

1. Left limit: $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} e^x = 1$
2. Right limit: $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x + 1) = 1$
3. Function value: $f(0) = e^0 = 1$
4. The three are equal, so the function is continuous at $x = 0$

Conclusion: this piecewise function is continuous at $x = 0$.

Types of Point of Discontinuity

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

1. Undefined points: the function is undefined at that point
2. Boundary discontinuities: discontinuous at the boundary of the domain
3. Composite function discontinuities: discontinuities caused by the discontinuity of the inner function

Determining Continuity

1. Basic exponential functions: continuous on $\mathbb{R}$
2. Composite exponential functions: the domain and the continuity of the inner function must be checked
3. Piecewise exponential 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 $f(x) = e^{\frac{1}{x}}$ at $x = 0$.

Exercise 2

Determine the continuity of $f(x) = e^{\sin x}$ on $\mathbb{R}$.

Exercise 3

Let $f(x) = \begin{cases} e^x, & x \leq 0 \\ x + 1, & x > 0 \end{cases}$. Determine the continuity of $f(x)$ at $x = 0$.

Exercise 4

Determine the continuity of $f(x) = 2^{\frac{1}{x-1}}$ at $x = 1$.

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Summary

Symbols Used in This Article