f is defined on [-5,5] as f(x)=`{{:("x if x is rational"),("-x if x is irrational"):}`

Options

A. f(x) is continuous at every x, except x =0
B. f(x) is discontinuous at every x, except x=0
C. f(x) is continuous everywhere
D. f(x) is discontinuous everywhere

Answer

To solve the problem, we need to analyze the function defined as follows:

\[ f(x) = \begin{cases} x & \text{if } x \text{ is rational} \\ -x & \text{if } x \text{ is irrational} \end{cases} \]

We will check the continuity of this function at different points in the interval \([-5, 5]\).

### Step 1: Check Continuity at \(x = 0\)

1. **Find the left-hand limit (LHL)** as \(x\) approaches \(0\):
- For rational \(x\), \(f(x) = x\).
- For irrational \(x\), \(f(x) = -x\).
- As \(x\) approaches \(0\) from the left, both rational and irrational numbers will approach \(0\).
- Thus,
\[ \lim_{x \to 0^-} f(x) = 0. \]

2. **Find the right-hand limit (RHL)** as \(x\) approaches \(0\):
- Similarly, as \(x\) approaches \(0\) from the right, both rational and irrational numbers will also approach \(0\).
- Thus,
\[ \lim_{x \to 0^+} f(x) = 0. \]

3. **Check the value of the function at \(x = 0\)**:
- Since \(0\) is rational, \(f(0) = 0\).

4. **Conclusion at \(x = 0\)**:
- Since \(LHL = RHL = f(0) = 0\), the function is continuous at \(x = 0\).

### Step 2: Check Continuity at \(x = 1\)

1. **Find the left-hand limit (LHL)** as \(x\) approaches \(1\):
- For rational \(x\), \(f(x) = x\) and as \(x\) approaches \(1\) from the left, \(f(x) \to 1\).
- Thus,
\[ \lim_{x \to 1^-} f(x) = 1. \]

2. **Find the right-hand limit (RHL)** as \(x\) approaches \(1\):
- For irrational \(x\), \(f(x) = -x\) and as \(x\) approaches \(1\) from the right, \(f(x) \to -1\).
- Thus,
\[ \lim_{x \to 1^+} f(x) = -1. \]

3. **Conclusion at \(x = 1\)**:
- Since \(LHL \neq RHL\), the function is discontinuous at \(x = 1\).

### Step 3: Check Continuity at Other Points

1. **General Case for \(x = a\) where \(a \neq 0\)**:
- For any rational \(a\):
- \(LHL = a\) (for rational) and \(RHL = -a\) (for irrational).
- For any irrational \(a\):
- \(LHL = -a\) (for irrational) and \(RHL = a\) (for rational).
- In both cases, \(LHL \neq RHL\).

2. **Conclusion for all \(x \neq 0\)**:
- The function is discontinuous at every point \(x \neq 0\).

### Final Conclusion

The function \(f(x)\) is continuous only at \(x = 0\) and discontinuous everywhere else in the interval \([-5, 5]\).

### Answer
The correct option is: **f of x is discontinuous at every x except x = 0.**

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