Understanding limits is an exciting adventure that helps us figure out how functions work, especially piecewise functions! Let’s jump right in!

What is Continuity?

A function is continuous if you can draw it without lifting your pencil from the paper. But how do we know for sure? That’s where limits come in!

Importance of Limits

Limits help us see how a function acts as it gets close to a certain point. For piecewise functions, which have different rules for different sections, we need to check the points where the sections meet.

Steps to Check Continuity in Piecewise Functions

1. Find the Points of Interest:

   • Look for the points where the pieces change. These are the key points we will focus on!

2. Calculate the Left-Hand Limit (LHL):

   • Find out what the function approaches as you get closer to the point from the left side. This is shown as:
     [ \lim_{x \to c^-} f(x) ]

3. Calculate the Right-Hand Limit (RHL):

   • Next, find out what the function approaches as you get closer to the point from the right side:
     [ \lim_{x \to c^+} f(x) ]

4. Check for Equality:

   • If the left-hand limit equals the right-hand limit (LHL = RHL), we can move on to check the function's value.

5. Evaluate the Function:

   • Lastly, figure out the value of the function at that point, ( f(c) ), to see if it matches the limits:
     [ f(c) = LHL = RHL ]

Conclusion

If all three conditions are met, great job! The function is continuous at that point! This method helps us not only with piecewise functions but also strengthens our overall understanding of continuity in math. Let’s keep learning more about limits and functions—it’s an awesome math journey!