Suppose that $a,b$ are real numbers, function $f(x) = ax+b$ satisfies $\mid f(x) \mid \leq 1$ for any $x \in [0,1]$. Find the range of values of $S= (a+1)(b+1).$
1. Given the function \( f(x) = ax + b \) and the condition \( |f(x)| \leq 1 \) for any \( x \in [0,1] \), we need to find the range of values of \( S = (a+1)(b+1) \).

2. Since \( f(x) \) is a linear function, its maximum and minimum values on the interval \([0,1]\) will occur at the endpoints \( x = 0 \) and \( x = 1 \).

3. Evaluating \( f(x) \) at the endpoints:
   \[
   f(0) = b \quad \text{and} \quad f(1) = a + b
   \]
   Given \( |f(x)| \leq 1 \), we have:
   \[
   |f(0)| = |b| \leq 1 \quad \text{and} \quad |f(1)| = |a + b| \leq 1
   \]

4. These inequalities can be written as:
   \[
   -1 \leq b \leq 1 \quad \text{and} \quad -1 \leq a + b \leq 1
   \]

5. To find the range of \( S = (a+1)(b+1) \), we need to express \( S \) in terms of \( a \) and \( b \) and then determine the possible values based on the constraints.

6. Expanding \( S \):
   \[
   S = (a+1)(b+1) = ab + a + b + 1
   \]

7. We need to find the maximum and minimum values of \( S \) under the given constraints. First, consider the constraints:
   \[
   -1 \leq b \leq 1 \quad \text{and} \quad -1 \leq a + b \leq 1
   \]

8. From \( -1 \leq a + b \leq 1 \), we can express \( a \) in terms of \( b \):
   \[
   -1 - b \leq a \leq 1 - b
   \]

9. To find the maximum value of \( S \), we need to consider the upper bounds of \( a \) and \( b \):
   - If \( b = 1 \), then \( -2 \leq a \leq 0 \). The maximum value of \( S \) occurs when \( a = 0 \):
     \[
     S = (0+1)(1+1) = 1 \cdot 2 = 2
     \]

   - If \( b = -1 \), then \( 0 \leq a \leq 2 \). The maximum value of \( S \) occurs when \( a = 2 \):
     \[
     S = (2+1)(-1+1) = 3 \cdot 0 = 0
     \]

   - If \( b = 0 \), then \( -1 \leq a \leq 1 \). The maximum value of \( S \) occurs when \( a = 1 \):
     \[
     S = (1+1)(0+1) = 2 \cdot 1 = 2
     \]

10. To find the minimum value of \( S \), we need to consider the lower bounds of \( a \) and \( b \):
    - If \( b = 1 \), then \( -2 \leq a \leq 0 \). The minimum value of \( S \) occurs when \( a = -2 \):
      \[
      S = (-2+1)(1+1) = -1 \cdot 2 = -2
      \]

    - If \( b = -1 \), then \( 0 \leq a \leq 2 \). The minimum value of \( S \) occurs when \( a = 0 \):
      \[
      S = (0+1)(-1+1) = 1 \cdot 0 = 0
      \]

    - If \( b = 0 \), then \( -1 \leq a \leq 1 \). The minimum value of \( S \) occurs when \( a = -1 \):
      \[
      S = (-1+1)(0+1) = 0 \cdot 1 = 0
      \]

11. Therefore, the range of values of \( S \) is from \(-2\) to \(2\).

The final answer is \(\boxed{[-2, 2]}\).