Are the following functions continuous at each point of their domains?
`e^(x)`

Answer

To determine if the function \( f(x) = e^x \) is continuous at each point of its domain, we will follow these steps:

### Step 1: Identify the Domain of the Function
The function \( f(x) = e^x \) is defined for all real numbers. Therefore, the domain of \( f \) is:
\[ \text{Domain} = (-\infty, \infty) \]

### Step 2: Understand the Definition of Continuity
A function is continuous at a point \( c \) if the following three conditions are satisfied:
1. \( f(c) \) is defined.
2. \( \lim_{x \to c} f(x) \) exists.
3. \( \lim_{x \to c} f(x) = f(c) \).

### Step 3: Check Continuity for Arbitrary Point \( c \)
Let \( c \) be any real number in the domain of \( f \).

1. **Check if \( f(c) \) is defined**:
Since \( f(x) = e^x \) is defined for all \( x \), \( f(c) = e^c \) is defined.

2. **Check if \( \lim_{x \to c} f(x) \) exists**:
The limit can be calculated as:
\[ \lim_{x \to c} f(x) = \lim_{x \to c} e^x = e^c \]
The limit exists and is equal to \( e^c \).

3. **Check if \( \lim_{x \to c} f(x) = f(c) \)**:
We have:
\[ \lim_{x \to c} f(x) = e^c = f(c) \]
Thus, this condition is satisfied.

### Step 4: Conclusion
Since all three conditions for continuity are satisfied for any arbitrary point \( c \) in the domain of \( f \), we conclude that the function \( f(x) = e^x \) is continuous at every point in its domain.

### Final Answer
The function \( f(x) = e^x \) is continuous at each point of its domain.

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