If `f:R rarrR` is a function f(x) = 5x , then f is

Options

A. continuous for every real x
B. discontinuous only at x = 0
C. discontinuous only at integral values of x
D. continuous only at x = 0

Answer

To determine the continuity of the function \( f(x) = 5x \), we will analyze the function step by step.

### Step 1: Understand the Definition of Continuity
A function \( f(x) \) is said to be 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 2: Identify the Function
The given function is \( f(x) = 5x \). This is a linear function.

### Step 3: Check for Breakpoints
Linear functions are continuous everywhere. To confirm this, we can check the conditions for continuity at any point \( c \) in the real numbers \( \mathbb{R} \).

### Step 4: Evaluate the Function at a Point
Let’s evaluate the function at a point \( c \):
- \( f(c) = 5c \) (This is defined for all \( c \in \mathbb{R} \)).

### Step 5: Evaluate the Limit
Next, we find the limit as \( x \) approaches \( c \):
- \( \lim_{x \to c} f(x) = \lim_{x \to c} 5x = 5c \).

### Step 6: Compare the Limit and Function Value
Now we compare the limit with the function value:
- \( \lim_{x \to c} f(x) = 5c = f(c) \).

Since both conditions are satisfied for any point \( c \), we conclude that \( f(x) \) is continuous everywhere.

### Conclusion
Thus, the function \( f(x) = 5x \) is continuous for every real \( x \).

### Final Answer
The correct option is: **Option 1: Continuous for every real \( x \)**.

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