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 \( f \) defined by

\[ f(x) = \begin{cases} \frac{x-4}{|x-4|} + a & \text{if } x < 4 \\ a + b & \text{if } x = 4 \\ \frac{x-4}{|x-4|} + b & \text{if } x > 4 \end{cases} \]

is continuous at \( x = 4 \), we need to ensure that the left-hand limit and the right-hand limit at \( x = 4 \) are equal to the function value at \( x = 4 \).

### Step 1: Calculate the left-hand limit as \( x \) approaches 4
For \( x < 4 \), we have:

\[ f(x) = \frac{x-4}{|x-4|} + a = \frac{x-4}{-(x-4)} + a = -1 + a \]

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 \( x > 4 \), we have:

\[ f(x) = \frac{x-4}{|x-4|} + b = \frac{x-4}{x-4} + b = 1 + b \]

Thus, the right-hand limit as \( x \) approaches 4 is:

\[ \lim_{x \to 4^+} f(x) = 1 + b \]

### Step 3: Set the limits equal to the function value at \( x = 4 \)
For the function to be continuous at \( x = 4 \), we need:

\[ \lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) = f(4) \]

This gives us the equations:

1. \( -1 + a = 1 + b \) (from left-hand and right-hand limits)
2. \( f(4) = a + b \)

### Step 4: Solve the equations
From the first equation:

\[ -1 + a = 1 + b \implies a - b = 2 \quad \text{(Equation 1)} \]

From the second equation, we have:

\[ a + b = a + b \quad \text{(Equation 2)} \]

### Step 5: Substitute and solve
We can express \( a \) in terms of \( b \) from Equation 1:

\[ a = b + 2 \]

Substituting \( a \) into Equation 2:

\[ (b + 2) + b = a + b \implies 2b + 2 = a + b \]

This gives us:

\[ 2b + 2 = (b + 2) + b \implies 2b + 2 = 2b + 2 \]

This is always true, so we can use Equation 1 to express \( a \) and \( b \) in terms of one variable.

### Step 6: Choose a value for \( b \)
Let’s choose \( b = 0 \):

Then from Equation 1:

\[ a = 0 + 2 = 2 \]

### Conclusion
Thus, the values of \( a \) and \( b \) that make the function continuous at \( x = 4 \) are:

\[ a = 2, \quad b = 0 \]