Identifying gaps in functions can be tough for 10th graders. Here are some common mistakes they make:

1. Forgetting What Continuity Means: Many students don’t remember that a function is continuous if:

   • The value at a point, $f(a)$, is defined.
   • The limit as you get close to that point, $\lim_{x \to a} f(x)$, exists.
   • The limit equals the value at that point: $\lim_{x \to a} f(x) = f(a)$.

2. Mixing Up Types of Discontinuity: Students often confuse removable, jump, and infinite discontinuities. This can lead to wrong answers.

3. Not Considering Piecewise Functions: They might miss discontinuities that come from the different rules used in piecewise functions.

4. Getting Limits Wrong: It's common for students to struggle with how to find limits, especially at points where there might be a discontinuity.

To help with these mistakes, teachers should focus on the clear definition of continuity.

Using visual aids like graphs can really help.

Plus, giving students plenty of practice with different kinds of functions is important.

Talking regularly about the role of limits in understanding functions can also strengthen their knowledge.