Let a function `f : R to R` be defined as `f(x)={{:(sinx-e^(x),if,xle0),(a+[-x],if,0ltxlt1),(2x-b,if,xge1):}`
Where [x] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:

Answer

To solve the problem, we need to ensure that the function \( f(x) \) is continuous at the points where the definition of the function changes, specifically at \( x = 0 \) and \( x = 1 \).

### Step 1: Check continuity at \( x = 1 \)

The function is defined as follows:
- For \( x < 0 \): \( f(x) = \sin x - e^x \)
- For \( 0 < x < 1 \): \( f(x) = a + [x] \) where \( [x] \) is the greatest integer less than or equal to \( x \)
- For \( x \geq 1 \): \( f(x) = 2x - b \)

To check continuity at \( x = 1 \), we need to find the left-hand limit, right-hand limit, and the function value at \( x = 1 \).

### Step 2: Calculate the left-hand limit as \( x \) approaches 1

The left-hand limit as \( x \) approaches 1 from the left is given by:
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (a + [x]) \]
Since \( [x] = 0 \) for \( x \) in the interval \( (0, 1) \):
\[ \lim_{x \to 1^-} f(x) = a + 0 = a \]

### Step 3: Calculate the right-hand limit as \( x \) approaches 1

The right-hand limit as \( x \) approaches 1 from the right is given by:
\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x - b) \]
Substituting \( x = 1 \):
\[ \lim_{x \to 1^+} f(x) = 2(1) - b = 2 - b \]

### Step 4: Set the limits equal for continuity

For \( f(x) \) to be continuous at \( x = 1 \), the left-hand limit must equal the right-hand limit:
\[ a = 2 - b \]
This is our first equation.

### Step 5: Check continuity at \( x = 0 \)

Next, we check the continuity at \( x = 0 \). The left-hand limit as \( x \) approaches 0 is:
\[ \lim_{x \to 0^-} f(x) = \sin(0) - e^0 = 0 - 1 = -1 \]

The value of the function at \( x = 0 \) is:
\[ f(0) = a + [0] = a + 0 = a \]

For continuity at \( x = 0 \):
\[ a = -1 \]

### Step 6: Substitute \( a \) into the first equation

Now we substitute \( a = -1 \) into the equation \( a = 2 - b \):
\[ -1 = 2 - b \]
Rearranging gives:
\[ b = 3 \]

### Step 7: Calculate \( a + b \)

Now we can find \( a + b \):
\[ a + b = -1 + 3 = 2 \]

### Final Answer

Thus, the value of \( a + b \) is \( \boxed{2} \).