Continuity of Exponential Functions

The exponential function is an important type of function in calculus; understanding its continuity properties is essential for studying more complicated types of function.

Basic Properties

The exponential function has the following basic properties:

• Monotonicity: the exponential function is a monotone function
• Continuity: continuous on its domain
• Inverse relation: the exponential function and the logarithmic function are inverses of each other
• Graph features: a smooth curve, with no jumps or breaks

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: $(0, +\infty)$

Proof of the Continuity of Exponential Functions

Proof Strategy

1. Use the monotonicity of the exponential function

   • The exponential function is monotone on $\mathbb{R}$

2. Use the properties of limits

   • For every $x_0 \in \mathbb{R}$, $\lim_{x \to x_0} a^x = a^{x_0}$

3. Prove continuity at an arbitrary point $x_0$

   • Since $\lim_{x \to x_0} f(x) = f(x_0)$
   • Therefore the function is continuous at $x_0$

Composite Exponential Functions

Example 1: $f(x) = e^{x^2}$

Analysis:

• The inner function $g(x) = x^2$ is continuous on $\mathbb{R}$
• The outer function $h(x) = e^x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = h(g(x))$ is continuous on $\mathbb{R}$

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

Analysis:

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

Example 3: $f(x) = e^{\frac{1}{x}}$

Analysis:

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

Graph Characteristics of Exponential Functions

Summary of Graph Features

1. Smoothness: the graph of the exponential function is a smooth curve, with no corners or breaks
2. Monotonicity: the exponential function is monotone on its domain
3. Asymptotes: there is a horizontal asymptote $y = 0$
4. Growth: the exponential function grows rapidly

Applications of Exponential Functions

1. Scientific Modelling

The exponential function is widely used in science:

• Population growth: describes the population growth model $P(t) = P_0 e^{rt}$
• Radioactive decay: describes the decay of radioactive substances $N(t) = N_0 e^{-\lambda t}$
• Chemical reactions: describe the rate of chemical reactions

2. Economic Applications

The exponential function has important applications in economics:

• Compound interest: describes the growth of capital $A = P e^{rt}$
• Inflation: describes changes in prices
• Economic growth: describes economic development

3. Engineering Applications

The exponential function has important applications in engineering:

• Signal processing: describes signal attenuation
• Circuit analysis: describes the charging and discharging of capacitors $V(t) = V_0 e^{-t/RC}$
• Heat conduction: describes temperature changes

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Exercises

Exercise 1

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

Exercise 2

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

Exercise 3

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

Exercise 4

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

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Summary

Symbols Used in This Article