Determine whether the following function is continuous at a. Use the continuity checklist to justify your answer.
\[
f(x)=\left\{\begin{array}{ll}
\frac{x^{2}-100}{x-10} & \text { if } x \neq 10 \\
8 & \text { if } x=10
\end{array} ; a=10\right.
\]
Select all that apply.
A. The function is continuous at $a=10$.
B. The function is not continuous at $a=10$ because $f(10)$ is undefined.
C. The function is not continuous at $a=10$ because $\lim _{x \rightarrow 10} f(x)$ does not exist.
D. The function is not cpntinuous at $a=10$ because $\lim _{x \rightarrow 10} f(x) \neq f(10)$.

Step 1 ：To determine whether the function is continuous at \(a=10\), we need to check three conditions: 1. \(f(a)\) is defined. 2. \(\lim_{x\rightarrow a} f(x)\) exists. 3. \(\lim_{x\rightarrow a} f(x) = f(a)\).

Step 2 ：From the function definition, we know that \(f(10) = 8\). So, the first condition is satisfied.

Step 3 ：To check the second and third conditions, we need to calculate the limit of \(f(x)\) as \(x\) approaches \(10\). We can simplify the function for \(x \neq 10\) as follows: \(f(x) = \frac{x^{2}-100}{x-10} = \frac{(x-10)(x+10)}{x-10} = x+10\).

Step 4 ：So, for \(x \neq 10\), \(f(x) = x+10\). Now, we can calculate the limit as \(x\) approaches \(10\). The limit of \(f(x)\) as \(x\) approaches \(10\) is \(20\). So, the second condition is satisfied.

Step 5 ：However, the third condition is not satisfied because \(\lim_{x\rightarrow 10} f(x) = 20 \neq f(10) = 8\).

Step 6 ：Final Answer: The correct answer is D. The function is not continuous at \(a=10\) because \(\lim_{x\rightarrow 10} f(x) \neq f(10)\). So, \(\boxed{\text{D}}\).