The fuction f(x) is defined in the interval [0,1] as follows:
`f(x)={:{(0", "x=0),(1/2-x", " 0ltxlt1/2),(1/2", " x=1/2),(2/3-x", " 1/2 lt x lt1),(1"," " "x=1):}`
Discuss the continuity of the function at `x=1/2`

Answer

To discuss the continuity of the function \( f(x) \) at \( x = \frac{1}{2} \), we need to check the following conditions:

1. The left-hand limit as \( x \) approaches \( \frac{1}{2} \).
2. The right-hand limit as \( x \) approaches \( \frac{1}{2} \).
3. The value of the function at \( x = \frac{1}{2} \).

### Step 1: Find the left-hand limit as \( x \) approaches \( \frac{1}{2} \)

For \( x < \frac{1}{2} \), the function is defined as:
\[ f(x) = \frac{1}{2} - x \]

Now, we calculate the left-hand limit:
\[ \lim_{x \to \frac{1}{2}^-} f(x) = \lim_{x \to \frac{1}{2}^-} \left(\frac{1}{2} - x\right) \]
Substituting \( x = \frac{1}{2} \):
\[ = \frac{1}{2} - \frac{1}{2} = 0 \]

### Step 2: Find the right-hand limit as \( x \) approaches \( \frac{1}{2} \)

For \( x > \frac{1}{2} \), the function is defined as:
\[ f(x) = \frac{2}{3} - x \]

Now, we calculate the right-hand limit:
\[ \lim_{x \to \frac{1}{2}^+} f(x) = \lim_{x \to \frac{1}{2}^+} \left(\frac{2}{3} - x\right) \]
Substituting \( x = \frac{1}{2} \):
\[ = \frac{2}{3} - \frac{1}{2} \]
To simplify \( \frac{2}{3} - \frac{1}{2} \), we find a common denominator (which is 6):
\[ = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \]

### Step 3: Find the value of the function at \( x = \frac{1}{2} \)

At \( x = \frac{1}{2} \), the function is defined as:
\[ f\left(\frac{1}{2}\right) = \frac{1}{2} \]

### Step 4: Check the continuity condition

For \( f(x) \) to be continuous at \( x = \frac{1}{2} \), the following must hold true:
\[ \lim_{x \to \frac{1}{2}^-} f(x) = \lim_{x \to \frac{1}{2}^+} f(x) = f\left(\frac{1}{2}\right) \]

From our calculations:
- Left-hand limit: \( 0 \)
- Right-hand limit: \( \frac{1}{6} \)
- Value of the function: \( \frac{1}{2} \)

Since \( 0 \neq \frac{1}{6} \) and \( 0 \neq \frac{1}{2} \), we conclude that:
\[ \text{Left-hand limit} \neq \text{Right-hand limit} \]

### Conclusion

The function \( f(x) \) is discontinuous at \( x = \frac{1}{2} \).

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