Understanding Continuity and Differentiability

A function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers or real numbers to real numbers. A function f is continuous on the closed interval [a, b] if f is continuous at every point in [a, b]. In this free online mathematics course, you will learn how to solve mathematical problems via functions and limits. Some students believed that if a limit exists, then, a function must be continuous, while others think that a limit and a function ought to be the same. Both of these beliefs are false since there are discontinuous functions whose limit exists at a point not equal to the function value. Learn how to ascertain the continuity, as well as the left-hand and right-hand limits of a function. You will be introduced to the derivative of a function at a given point, the essence of differentiation relating to functions and how to differentiate composite functions using the chain rule.

Next, you will discern the derivatives of inverse trigonometric and logarithmic functions and how to work successfully and efficiently with function notation and substitution. The analytical and numerical methods for interpreting and representing the definition of a derivative will be shown. The next topics include the derivative of a function with respect to x at a generic x-value, how to define continuity in terms of limits, the systems for evaluating the limits of a continuous function, and methods for identifying the points of discontinuity from an algebraic description of a function. You will then discuss how to formulate a limit statement that reveals the continuity status of a given function at a given point,  how to apply the laws of algebraic limits like scalar multiplication, quotient, substitution, etc., to compute the limits of algebraic combinations. The use of different alphabetical symbols will help you get a more complete idea of mathematical concepts because it is a powerful resource that will assist you when solving mathematical problems. The last section of content teaches you to recognize, represent, and translate the concept of continuity in its different representations, how to perform algebraic manipulations, as well as how to enable the computation of certain indeterminate forms arising from limits of different quotients. Similarly, you will learn to discern if an elementary function admits a continuous extension to boundary points of the domain and then the criteria for the continuity of a function at a point will be discussed. This forms the basis for recognizing the discontinuities of algebraic and geometric functions. What is the connectivity between the limits and continuity of a function? What are the derivatives of trigonometric and inverse trigonometric functions? This mathematics course will examine these subjects and consider the facts affecting and surrounding them. 

Whether you are studying, teaching or need to apply complex mathematics in your daily work, this mathematics course will teach you advanced skills for solving mathematical problems that involve continuity and differentiability and enrolling in this free online course will unlock a whole new world of advanced mathematics.