Examine the following functions for continuity :
`f(x) = x - 5`

Answer

To determine whether the function \( f(x) = x - 5 \) is continuous, we will follow the definition of continuity at a point.

### Step-by-Step Solution:

1. **Identify the function**: We have the function \( f(x) = x - 5 \). This is a linear function, which is a type of polynomial function.

2. **Recall the definition of continuity**: A function \( f \) is continuous at a point \( a \) if:
\[ \lim_{x \to a} f(x) = f(a) \]
This means that the limit of the function as \( x \) approaches \( a \) must equal the value of the function at \( a \).

3. **Choose a point \( a \)**: Let \( a \) be any real number. Since \( f(x) \) is defined for all real numbers, we can choose any \( a \).

4. **Calculate \( f(a) \)**:
\[ f(a) = a - 5 \]

5. **Calculate the limit \( \lim_{x \to a} f(x) \)**:
\[ \lim_{x \to a} f(x) = \lim_{x \to a} (x - 5) = a - 5 \]

6. **Compare \( f(a) \) and \( \lim_{x \to a} f(x) \)**:
We have:
\[ f(a) = a - 5 \quad \text{and} \quad \lim_{x \to a} f(x) = a - 5 \]
Since both are equal, we have:
\[ f(a) = \lim_{x \to a} f(x) \]

7. **Conclusion**: Since \( f(a) = \lim_{x \to a} f(x) \) holds for any real number \( a \), we conclude that the function \( f(x) = x - 5 \) is continuous for all real numbers.

### Final Statement:
Thus, the function \( f(x) = x - 5 \) is continuous everywhere on the real number line.

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