Continuity

The video begins with an introduction to the concept of continuity in functions. The traditional definition is discussed, which involves the ability to draw a function's graph without lifting the pencil. The presenter then introduces a new definition from calculus, stating that a function is continuous at a specific point if the limit as x approaches that point equals the function's value at that point. The concept of discontinuity is also explained, with examples of 'holes' in the graph and undefined function values at certain points. Two types of discontinuities are highlighted: removable and non-removable, with the former being like a point removed from an otherwise continuous graph and the latter involving jumps or vertical asymptotes where the limit does not exist.

This section of the video focuses on identifying when a function is discontinuous and classifying the type of discontinuity. Examples are provided, including a function with a vertical asymptote at x = -3 due to division by zero, which is a non-removable discontinuity. Another example is a piecewise function with a removable discontinuity at x = 1, where the function is defined differently on either side of the point. The video also covers rational functions with redundant factors that, when canceled, create a hole in the graph, indicating a removable discontinuity. The presenter emphasizes understanding the visual representation of these discontinuities rather than their mathematical definitions.

The video moves on to discuss continuity in piecewise functions, illustrating how to determine the values of constants that make the function continuous at certain points. An example is given where a quadratic function needs to meet a linear function at x = 2, and algebraic methods are used to find the value of k ensuring continuity. The concept of the intermediate value theorem is introduced as the first existence theorem, stating that if a function is continuous on a closed interval and a value lies between the function's values at the endpoints of the interval, then the function must cross that value. The presenter notes that the theorem is taken as self-evident and not proven within the scope of the class.

The presenter explores the minimum number of x-intercepts a continuous function can have, using both graphical and numerical methods. A sketch illustrates that at least three x-intercepts are possible by connecting five points in a minimal way. The concept of sign changes in a table of values is also discussed as evidence of x-intercepts. The video then poses a question regarding the values of b that ensure two solutions to an equation involving a continuous function on a given interval, concluding that b must be less than one to guarantee two solutions.

The final part of the video introduces the concept of the average rate of change, equated to the familiar 'rise over run' from earlier mathematics. The formula for average rate of change is presented as f(b) - f(a) / (b - a), and its interpretation as the slope of a secant line between two points on the graph of y = f(x) is explained. The units of rate of change are discussed, and an example involving the rate of water pumped into a tank is used to demonstrate the calculation of average rate of change with units. The video concludes with a more complex example involving the sine function to practice calculating the average rate of change over a specific interval.