Great question! Understanding the difference between checking differentiability on closed versus open intervals is key to applying calculus theorems correctly. Let's break down the concepts:

Differentiability on Open vs. Closed Intervals

1. Differentiability on an Open Interval $(a, b)$:

   A function $f(x)$ is said to be differentiable on an open interval $(a, b)$ if the derivative $f'(x)$ exists for every point $x$ in that interval.

   • Open Interval $(a, b)$: An interval that does not include its endpoints $a$ and $b$. In other words, $a < x < b$.
   • Checking Differentiability on $(a, b)$: To check differentiability on an open interval, you only consider points strictly between $a$ and $b$. You do not need to consider the behavior of the function at the endpoints $a$ and $b$.

   Example: If $f(x) = x^2$, then $f'(x) = 2x$ exists for every $x$ in the interval $(-1, 1)$. Thus, $f(x)$ is differentiable on $(-1, 1)$.

2. Differentiability on a Closed Interval $[a, b]$:

   A function $f(x)$ is said to be differentiable on a closed interval $[a, b]$ if the derivative $f'(x)$ exists for every point in the open interval $(a, b)$ and also if the derivative exists at the endpoints $a$ and $b$ as one-sided limits.

   • Closed Interval $[a, b]$: An interval that includes its endpoints $a$ and $b$. In other words, $a \leq x \leq b$.
   • Checking Differentiability on $[a, b]$: To check differentiability on a closed interval, you must check:
     • The function is differentiable on the open interval $(a, b)$.
     • The derivative $f'(x)$ exists as a right-hand derivative at $x = a$ and as a left-hand derivative at $x = b$.

   Example: If $f(x) = x^2$, then $f'(x) = 2x$ exists for every $x$ in the interval $[0, 1]$ and also at the endpoints $x = 0$ and $x = 1$ as one-sided derivatives:

   • Right-hand derivative at $x = 0$: $\lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h^2}{h} = 0$
   • Left-hand derivative at $x = 1$: $\lim_{h \to 0^-} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0^-} \frac{(1+h)^2 - 1}{h} = \lim_{h \to 0^-} \frac{2h + h^2}{h} = 2$

Key Differences Between Open and Closed Intervals for Differentiability

• Open Interval $(a, b)$: Only considers differentiability for points within the interval, not at the endpoints.
• Closed Interval $[a, b]$: Requires checking differentiability at all points within the interval and ensuring that the one-sided derivatives exist at the endpoints.

Why This Matters for Calculus Theorems

• Rolle's Theorem and LMVT: These theorems specifically require that the function be differentiable on the open interval $(a, b)$ and continuous on the closed interval $[a, b]$. This is because the derivative must exist strictly between $a$ and $b$, but the behavior at the endpoints doesn't directly affect the result as long as the function is continuous there.

• Checking Differentiability: In practical terms, checking differentiability on $(a, b)$ is often easier because you don’t need to worry about the boundary behavior. When checking differentiability on $[a, b]$, you need to carefully evaluate the one-sided limits at the endpoints, which can sometimes introduce additional complexity.

Tip: Always ensure you understand whether a theorem or a problem is asking for differentiability on an open or closed interval, as this will determine how you check the function's behavior at the endpoints.