Find the values of a and b sucht that the function f defined by
`fx = {{:((x-4)/(|x-4|)+a, if x lt 4),(a+b,if x =4),((x-4)/(|x-4|)+b, if x gt 4):}`
is a continous function at `x = 4` .

Explanation

To find the values of \( a \) and \( b \) such that the function \( f \) is continuous at \( x = 4 \), we need to ensure that the left-hand limit, the right-hand limit, and the function value at \( x = 4 \) are all equal.

The function is defined as follows:
\[ f(x) = \begin{cases} \frac{x-4}{|x-4|} + a & \text{if } x < 4 \\ a + b & \text{if } x = 4 \\ \frac{x-4}{|x-4|} + b & \text{if } x > 4 \end{cases} \]

### Step 1: Calculate the Left-Hand Limit as \( x \) approaches 4
For \( x < 4 \):
\[ f(x) = \frac{x-4}{|x-4|} + a = \frac{x-4}{-(x-4)} + a = -1 + a \]
Thus, the left-hand limit as \( x \) approaches 4 is:
\[ \lim_{x \to 4^-} f(x) = -1 + a \]

### Step 2: Calculate the Right-Hand Limit as \( x \) approaches 4
For \( x > 4 \):
\[ f(x) = \frac{x-4}{|x-4|} + b = \frac{x-4}{x-4} + b = 1 + b \]
Thus, the right-hand limit as \( x \) approaches 4 is:
\[ \lim_{x \to 4^+} f(x) = 1 + b \]

### Step 3: Set Up the Continuity Condition
For the function to be continuous at \( x = 4 \), the left-hand limit, right-hand limit, and the function value at \( x = 4 \) must all be equal:
\[ -1 + a = a + b = 1 + b \]

### Step 4: Solve the First Equation
From the first part of the equality:
\[ -1 + a = a + b \]
Simplifying gives:
\[ -1 = b \quad \text{(1)} \]

### Step 5: Solve the Second Equation
From the second part of the equality:
\[ a + b = 1 + b \]
Simplifying gives:
\[ a = 1 \quad \text{(2)} \]

### Step 6: Substitute to Find Values
From equation (1), we have \( b = -1 \). From equation (2), we have \( a = 1 \).

### Conclusion
Thus, the values of \( a \) and \( b \) that make the function continuous at \( x = 4 \) are:
\[ \boxed{a = 1, b = -1} \]