Determine whether the following function is continuous at a. Use the continuity checklist to justify your answer.
$$f(x)=\{\begin{array}{ll}\frac{x^2-100}{x-10}& \text{ if }x\ne 10\\ 8& \text{ if }x=10\end{array};a=10$$
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 $\underset{x\to 10}{lim}f(x)$ does not exist.
D. The function is not cpntinuous at $a=10$ because $\underset{x\to 10}{lim}f(x)\ne f(10)$.

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

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 $x\ne 10$ as follows: $f(x)=\frac{x^2-100}{x-10}=\frac{(x-10)(x+10)}{x-10}=x+10$.

Step 4 ：So, for $x\ne 10$, $f(x)=x+10$. 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 $\underset{x\to 10}{lim}f(x)=20\ne f(10)=8$.

Step 6 ：Final Answer: The correct answer is D. The function is not continuous at $a=10$ because $\underset{x\to 10}{lim}f(x)\ne f(10)$. So, $\boxed{{\displaystyle \text{D}}}$.