A graph for a function that's smooth without any holes, jumps, or asymptotes is called continuous. Your pre-calculus teacher will tell you that three things have to be true for a function to be …

May 5, 2025 · Determine the domain: The function is defined for all real x such that x \neq 1 and x \neq -1. Conclude continuity on the domain: Because rational functions are continuous on their …

Dec 21, 2020 · Using the definition, determine whether the function \(f(x)=\begin{cases}2x+1 & if x<1\\2 & if x=1\\ −x+4 & if x>1\end{cases}\) is continuous at \(x=1\). If the function is not …

Aug 8, 2024 · We can prove the continuity of the function using the concept of limits as the limit of any function tells us where the function approaches when input approaches some real number.

We begin our investigation of continuity by exploring what it means for a function to have continuity at a point. Intuitively, a function is continuous at a particular point if there is no break …

Look out for holes, jumps or vertical asymptotes (where the function heads up/down towards infinity). Try these different functions so you get the idea: (Use slider to zoom, drag graph to …

Sep 27, 2023 · Checking the continuity of a function is easy! The simple rule for checking is tracing your pen on the curve. If you have to pick up your pen, the function is discontinuous. …

There are several theorems on a continuous function. Here are the most important theorems. Theorem 1: All polynomial functions are continuous on (-∞, ∞). Theorem 2: The functions e x, …

Nov 16, 2022 · Example 1 Given the graph of f (x) f (x), shown below, determine if f (x) f (x) is continuous at x =−2 x = − 2, x =0 x = 0, and x = 3 x = 3. To answer the question for each point …

How to determine if a function is continuous? In this section, we’ll discuss the more formal conditions a function must satisfy before we can establish that it’s continuous throughout its …