If $f$ is differentiable at $x=a$, which of the following could be false?
(A) fis continuous at $x=a$.
(B) $\lim _{x \rightarrow a} f(x)$ exists.
(C) $\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$ exists.
(D) $f^{\prime}(a)$ is defined.
(E) $f^{\prime \prime}(a)$ is defined.

Step 1 ：The definition of a function being differentiable at a point \(x=a\) is that the limit \(\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\) exists, which is the derivative of the function at that point.

Step 2 ：(A) If a function is differentiable at a point, it is also continuous at that point. This is because differentiability at a point implies the existence of a limit at that point, which is a requirement for continuity. So, this statement is true.

Step 3 ：(B) As mentioned above, differentiability at a point implies the existence of a limit at that point. So, this statement is also true.

Step 4 ：(C) This is the definition of differentiability at a point. So, this statement is true.

Step 5 ：(D) This is also a part of the definition of differentiability. The derivative of the function at the point \(x=a\) is defined as \(\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\). So, this statement is true.

Step 6 ：(E) However, the second derivative of the function at the point \(x=a\), \(f^{\prime \prime}(a)\), is not necessarily defined. The function being differentiable at a point does not guarantee that its second derivative at that point exists. So, this statement could be false.

Step 7 ：Therefore, the answer is (E).