`f(x)= {{:((|x-4|)/(2(x-4)), if x ne 4),(0,if x = 4):}` at `x = 4`.

Explanation

To determine whether the function \( f(x) \) is continuous at \( x = 4 \), we need to check the following:

1. The value of the function at \( x = 4 \).
2. The left-hand limit as \( x \) approaches 4.
3. The right-hand limit as \( x \) approaches 4.
4. If both limits are equal to the value of the function at \( x = 4 \).

Given the function:
\[ f(x) = \begin{cases} \frac{|x-4|}{2(x-4)} & \text{if } x \neq 4 \\ 0 & \text{if } x = 4 \end{cases} \]

### Step 1: Find \( f(4) \)
From the definition of the function, we have:
\[ f(4) = 0 \]

### Step 2: Find the left-hand limit as \( x \) approaches 4
To find the left-hand limit, we compute:
\[ \lim_{x \to 4^-} f(x) = \lim_{x \to 4^-} \frac{|x-4|}{2(x-4)} \]
Since \( x \) is approaching 4 from the left, \( |x-4| = -(x-4) = 4-x \). Thus, we can rewrite the limit:
\[ \lim_{x \to 4^-} \frac{4-x}{2(x-4)} = \lim_{x \to 4^-} \frac{4-x}{2(-1)(4-x)} = \lim_{x \to 4^-} \frac{1}{-2} = -\frac{1}{2} \]

### Step 3: Find the right-hand limit as \( x \) approaches 4
Now, we compute the right-hand limit:
\[ \lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} \frac{|x-4|}{2(x-4)} \]
Since \( x \) is approaching 4 from the right, \( |x-4| = x-4 \). Thus, we can rewrite the limit:
\[ \lim_{x \to 4^+} \frac{x-4}{2(x-4)} = \lim_{x \to 4^+} \frac{1}{2} = \frac{1}{2} \]

### Step 4: Compare the limits and the function value
Now we compare the left-hand limit, right-hand limit, and the function value:
- Left-hand limit: \( -\frac{1}{2} \)
- Right-hand limit: \( \frac{1}{2} \)
- Function value at \( x = 4 \): \( f(4) = 0 \)

Since the left-hand limit and right-hand limit are not equal, and neither of them equals \( f(4) \), we conclude that:

### Conclusion
The function \( f(x) \) is not continuous at \( x = 4 \).

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