For any real number c, we have limx→ccosx = cosc and cosx is defined for all real numbers. Thus, the function cosx is continuous everywhere. Now, we will prove that cosx is continuous for all values of x by the epsilon-delta method. We will use the following two formulas: Prove cosx is continuous Let f(x)=cosx … Read more
The modulus of x, that is, |x| is a continuous function. In this post, we will learn how to prove mod x is continuous. Let us now show that mod x is continuous at the point x=0. f(x)=|x| is continuous at 0 Let f(x)=|x|. Note that we have So we obtain that $\lim\limits_{x \to 0} … Read more
In this post, we will show that the trigonometric function sinx is continuous for all values of x. Here, we will use the limit method as well as the epsilon-delta definition. Note that $\lim\limits_{x \to c} \sin x=\sin c$ and $\sin x$ is defined for all real numbers. Thus, we can say that the function … Read more
In this post, we will learn about the epsilon-delta definition of continuity with solved examples. To learn this, let us first recall the definition of continuity. Definition of Continuity: A real-valued function f(x) is said to be continuous at a point x=a in the domain of f(x) if the following condition is satisfied: $\lim\limits_{x \to … Read more