Find all Values of c so that the Piecewise Function is Continuous

TL;DR

Find values of C that make a given function continuous.

Transcript

hey what's up YouTube this is a pretty interesting problem says find all values of C so that the function is continuous on the entire real line so first it's worth noting that 1 minus x squared on its own is a continuous function likewise X is also continuous but when you put them together like this there might be a discontinuity at C so the questi... Read More

Key Insights

• ❓ A function can be discontinuous when two continuous functions are combined.
• 😥 Continuity at a specific point can be ensured by making one-sided limits equal to each other.
• 🎅 In the given problem, the equation 1 - C^2 = C is solved using the quadratic formula.
• 🫥 By solving the equation, the values of C that make the function continuous on the entire real line are obtained.
• 🥡 Taking one-sided limits is a technique to determine the continuity of a function.
• 👔 The quadratic formula is helpful in solving equations with the form ax^2 + bx + c = 0.
• ↔️ The one-sided limits for the given function are 1 - C^2 (from the left) and C (from the right).

Questions & Answers

Q: What does it mean for a function to be continuous at a certain point?

A function is continuous at a point when the limit as x approaches that point of f(x) equals f of the point.

Q: How can one force a limit to exist and make a function continuous?

By taking one-sided limits and setting them equal to each other, the limit at the point can be made to exist, ensuring continuity.

Q: What are the one-sided limits in this problem?

The limit from the left (as x approaches C from the left) is 1 - C^2, and the limit from the right (as x approaches C from the right) is C.

Q: How is the equation 1 - C^2 = C solved?

By rearranging the equation and using the quadratic formula (with X substituted by C), the values of C can be determined.

Summary & Key Takeaways

• A function is continuous if the limit as x approaches C of f(x) is equal to f(C).

• By taking one-sided limits and setting them equal to each other, the function's limit at C can be forced to exist, ensuring continuity.

• The equation 1 - C^2 = C is solved to find the values of C that make the function continuous.