Continuity of a Piecewise Function on the Entire Real Line
In mathematics, the concept of continuity plays a crucial role in understanding the behavior of functions. A function is said to be continuous if it does not have any abrupt changes or jumps in its values. In this article, we will explore the continuity of a specific piecewise function on the entire real line. The given function is defined as follows: \[ f(x)=\left\{\begin{array}{l}x+2, x \leqslant 0 . \\ \frac{\sin 2 x}{x}, x >0\end{array}\right. \] To determine the continuity of this function, we need to analyze its behavior at the points of transition, which in this case is at \( x = 0 \). Let's first consider the left-hand limit as \( x \) approaches 0. As \( x \) approaches 0 from the left side, the function \( f(x) = x + 2 \) is defined. Since this is a linear function, it is continuous for all values of \( x \). Therefore, the left-hand limit of the function as \( x \) approaches 0 exists and is equal to 2. Next, let's examine the right-hand limit as \( x \) approaches 0. As \( x \) approaches 0 from the right side, the function \( f(x) = \frac{\sin 2x}{x} \) is defined. The sine function is also continuous for all values of \( x \), except at \( x = 0 \) where it is undefined. However, we can use the fact that \( \sin x \) is bounded between -1 and 1 to show that the right-hand limit of the function as \( x \) approaches 0 exists and is equal to 0. Since the left-hand limit and the right-hand limit of the function as \( x \) approaches 0 exist and are equal, we can conclude that the function is continuous at \( x = 0 \). Now, let's consider the behavior of the function for values of \( x \) other than 0. For \( x \leq 0 \), the function \( f(x) = x + 2 \) is a linear function, which is continuous for all values of \( x \). For \( x > 0 \), the function \( f(x) = \frac{\sin 2x}{x} \) is a composition of continuous functions, and therefore, it is also continuous for all values of \( x \) except at \( x = 0 \). In conclusion, the given piecewise function is continuous on the entire real line, \( (-\infty, \infty) \), except at the point \( x = 0 \) where it has a removable discontinuity. This means that the function can be made continuous at \( x = 0 \) by redefining its value at that point. Overall, understanding the continuity of functions is essential in analyzing their behavior and properties. By examining the behavior of a piecewise function at points of transition, we can determine its continuity on specific intervals or the entire real line.