Continuity and Limits in Functions

The video tutorial introduces the concept of continuity in real functions, leading up to calculus. It explains continuity informally as the ability to draw a graph without lifting a pen and formally using limits and notation. The tutorial provides examples, such as the function 1/x, to illustrate continuity and discontinuity, emphasizing the importance of approaching values from both directions to determine continuity.

Which of the following is an example of a continuous function?

• A piecewise function

• A function with a hole

• A straight line

• A function with a jump discontinuity

What is the informal definition of a continuous function at a point?

• A function that can be drawn without lifting the pen

• A function that has a hole at the point

• A function that is not defined at the point

• A function that has a jump at the point

According to the formal definition, a function is continuous at a point if:

• The function has a hole at that point

• The function has a jump at that point

• The function is not defined at that point

• The limit from the left equals the limit from the right and both equal the function value at that point

What does the formal definition of continuity use to determine if a function is continuous at a point?

• Asymptotes

• Limits

• Integrals

• Derivatives

When testing the continuity of f(x) = 1/x at x = 1, what is the limit as x approaches 1 from the left?

• Negative infinity

• Infinity

• 1

• 0

For the function f(x) = 1/x, what is the function value at x = 1?

• 0

• 1

• Infinity

• Undefined

Why is f(x) = 1/x not continuous at x = 0?

• The function is not defined at x = 0

• The limits from the left and right are equal

• The function value at x = 0 is 0

• The function has a jump at x = 0

What happens to the limit of f(x) = 1/x as x approaches 0 from the negative side?

• It approaches 0

• It approaches 1

• It approaches negative infinity

• It remains constant

What is the behavior of the limit of f(x) = 1/x as x approaches 0 from the positive side?

• It approaches 1

• It remains constant

• It approaches 0

• It approaches infinity

In the context of limits, what does it mean for a value to be 'approached'?

• The value is exactly reached

• The value is ignored

• The value is never reached but can be made arbitrarily close

• The value is surpassed