Let f be a thrice differentiable function with continuous third derivative on R.

Steps to Solve

1. State the properties of the function
   Given that $f$ is a thrice differentiable function with a continuous third derivative, we can denote the derivatives as follows:

   • First derivative: $f'(x)$
   • Second derivative: $f''(x)$
   • Third derivative: $f'''(x)$

2. Utilize Taylor's Theorem
   Using Taylor's Theorem, we can express $f(x)$ around a point $c$ as:
   $$ f(x) = f(c) + f'(c)(x - c) + \frac{f''(c)}{2}(x-c)^2 + \frac{f'''(c)}{6}(x-c)^3 + R_3(x) $$
   where $R_3(x)$ is the remainder term defined as:
   $$ R_3(x) = \frac{f^{(4)}(\xi)}{24}(x-c)^4 $$ for some $\xi$ between $x$ and $c$.

3. Understand implications of the continuity
   Since $f'''(x)$ is continuous, we can conclude that it is bounded on any closed interval of $\mathbb{R}$, which provides significant information about the behavior of $f(x)$ and its derivatives.

4. Explore Mean Value Theorem
   The Mean Value Theorem can be applied multiple times. Between two points $a$ and $b$, there exists a point $c$ such that:
   $$ f'(b) - f'(a) = f''(c)(b - a) $$
   and
   $$ f''(b) - f''(a) = f'''(d)(b - a) $$
   for some $c$ and $d$ in the interval $[a, b]$.