Calculus 1 Lecture 1.4: Continuity of Functions

The first paragraph introduces the concept of continuity in functions. It explains that a function is considered continuous if it has no gaps, breaks, or asymptotes, meaning you can draw its graph without lifting the pencil. The mathematical definition of continuity involves three conditions: the function must be defined at a point, the limit at that point must exist, and the function's value at the point must equal the limit. This paragraph also touches on the idea of continuity over a range of points and the concept of removable discontinuities, which are gaps that can be filled with a single point.

The second paragraph delves into the application of the continuity definition with visual examples. It discusses how to check for continuity at a point by ensuring the existence of the function value, the limit, and that the limit equals the function value at that point. The paragraph also explains the concept of removable discontinuities, where a hole in the function can be filled with a single point, and non-removable discontinuities, which are like jumps that cannot be filled.

The third paragraph focuses on the nuances of continuity at endpoints of a function's domain. It clarifies that while continuity within an interval can be checked by ensuring the function's existence and the equality of the function value to the limit, endpoints require checking one-sided limits. The paragraph emphasizes that if a function is continuous at every point within an interval, it is considered continuous on that interval. However, the endpoints are only included if they satisfy the conditions of continuity from the appropriate side.

The fourth paragraph outlines the process of proving continuity over a closed interval. It involves checking the function's continuity on the open interval between the endpoints and then verifying continuity at each endpoint separately. The paragraph explains that this requires checking one-sided limits at the endpoints and ensuring they match the function's value at those points. The process is broken down into three parts: checking the open interval, the right endpoint, and the left endpoint.

The fifth paragraph explores the properties of limits and continuity in the context of composite functions. It establishes that if two functions are continuous at a point, their sum, difference, and product are also continuous at that point. However, division requires the additional condition that the denominator is not zero at that point to ensure continuity. The paragraph also touches on the concept of one-sided limits and how they relate to continuity at endpoints.

The sixth paragraph discusses the relationship between polynomial functions and continuity. It states that since polynomials can be evaluated by simply plugging in the value of x (due to their being composed of terms with non-negative integer exponents), every polynomial function is continuous everywhere. This leads to the conclusion that rational functions, which are ratios of polynomials, are also continuous everywhere except where the denominator equals zero, which would result in a discontinuity.

The seventh paragraph provides a method for identifying discontinuities in rational functions. It explains that discontinuities occur where the denominator equals zero and that factoring the numerator and denominator can help determine the nature of the discontinuity. If a discontinuity can be 'crossed out' by factoring, it is a removable discontinuity (a hole). If not, it is an asymptote. The paragraph also emphasizes that one cannot simply eliminate discontinuities without proper justification.

The eighth paragraph focuses on proving the continuity of the absolute value function. It breaks down the absolute value function into two cases (for x greater than zero and for x less than zero) and shows that at the point where x equals zero, the one-sided limits exist and are equal. This fulfills the criteria for continuity, proving that the absolute value function is continuous everywhere.

The ninth paragraph discusses the continuity of compositions and inverse functions. It states that if two functions are continuous everywhere, their composition will also be continuous everywhere. Additionally, if a function is continuous on its domain, its inverse will be continuous on its domain, with the caveat that the domain of the inverse function is the range of the original function.

The tenth paragraph introduces the Intermediate Value Theorem, which states that if a function is continuous on a closed interval and takes on different signs at the endpoints, then there exists at least one root within that interval. The theorem is particularly useful for approximating roots of functions, as it guarantees the existence of a point that maps to a given value within a specified range of the function's outputs.

The eleventh paragraph provides a practical application of the Intermediate Value Theorem for approximating roots of functions. It describes a method for narrowing down the location of a root by evaluating the function at intervals and observing the sign changes of the results. This iterative process allows for increasingly accurate approximations of the root's location.

The twelfth paragraph offers a manual approach to finding roots without the aid of calculators or computers. It involves creating a table of values and iteratively refining the interval within which a root is known to exist based on the function's output signs. This method, while time-consuming, can achieve a high degree of accuracy and provides a deeper understanding of the function's behavior.