A function `f : R to R` is defined as follows : f(x)=`{{:(x","if x le 1),(5","if x > 1):}`
Which one of the following is true ?

Options

A. f is continuous at 0 and 1
B. f is continuous at 1 and 2
C. f is continuous at 0 and 2
D. f is continuous at 0,1 and 2

Answer

To determine the continuity of the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} x & \text{if } x \leq 1 \\ 5 & \text{if } x > 1 \end{cases} \]

we need to analyze the function at different points, particularly at the point where the definition of the function changes, which is \( x = 1 \).

### Step 1: Evaluate \( f(1) \)

Since \( 1 \) is included in the first case of the piecewise function, we have:

\[ f(1) = 1 \]

### Step 2: Evaluate the left-hand limit as \( x \) approaches \( 1 \)

We need to find the limit of \( f(x) \) as \( x \) approaches \( 1 \) from the left:

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x = 1 \]

### Step 3: Evaluate the right-hand limit as \( x \) approaches \( 1 \)

Next, we find the limit of \( f(x) \) as \( x \) approaches \( 1 \) from the right:

\[ \lim_{x \to 1^+} f(x) = 5 \]

### Step 4: Check for continuity at \( x = 1 \)

For the function \( f(x) \) to be continuous at \( x = 1 \), the following must hold:

\[ \lim_{x \to 1} f(x) = f(1) \]

However, we found:

- \( f(1) = 1 \)
- \( \lim_{x \to 1^-} f(x) = 1 \)
- \( \lim_{x \to 1^+} f(x) = 5 \)

Since the left-hand limit and the right-hand limit are not equal (\( 1 \neq 5 \)), the limit does not exist at \( x = 1 \).

### Conclusion

Thus, the function \( f(x) \) is discontinuous at \( x = 1 \).

### Final Answer

The function \( f(x) \) is discontinuous at \( x = 1 \).

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