Question

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\left\\{\begin{array}{ll}3+x, & x \leq 2 \\ x^{2}+1, & x>2\end{array}\right.\)

Short answer

The function is continuous on all real numbers. This is because both pieces of the function are continuous on their defined intervals, and at the point where they transition (\(x = 2\)), the left-hand limit, the function value, and the right-hand limit all equal 5.

Step-by-step solution

Identify the rules and the intervals

Our function is defined as \(f(x) = 3+x\) when \(x \leq 2\), and \(f(x) = x^2 + 1\) when \(x > 2\). So, we have two intervals to analyze, \(-\infty < x \leq 2\) and \(2 < x < \infty\).

Determine the continuity of the individual functions

The functions \(3+x\) and \(x^2 + 1\) are both polynomials, which are continuous everywhere. This means they are continuous on their respective defined intervals.

Check continuity at transition point

The one point we still need to consider is at \(x = 2\), where the two parts of our function meet. The limit from the left (when \(x = 2-\)) is \(2 + 3 = 5\). The function's value at \(x = 2\) is also \(2 + 3 = 5\). The limit from the right (when \(x = 2+\)) is \(2^2 + 1 = 5\). Since these three values are all equal, the function is also continuous at \(x = 2\).

Key concepts

Discontinuity

When studying function continuity, it's vital to understand what makes a function discontinuous. A discontinuity occurs when there's a sudden break or jump in the graph of a function. It's where the function isn't smooth or uninterrupted. For a function to be continuous at a point x = a, three conditions must be met:

• The function must be defined at a, meaning f(a) exists.
• The limit as x approaches a must exist.
• The limit of the function as x approaches a must equal f(a).

If any of these conditions are not satisfied, the function will have a discontinuity at x = a. Discontinuities can be classified as jump, point, or essential. In the given step-by-step solution, the continuity of f(x) at x = 2 is scrutinized to confirm there's no discontinuity since all the conditions are fulfilled at this transition point.

Limits

Limits are a fundamental concept in calculus and analysis, concerning the behavior of functions as they approach a certain point. They describe the value that a function 'tends to' as the input 'tends to' a particular value. For continuous functions, the limit at a point x = a is simply the value of the function at a: \( \text{{lim}}_{x \to a} f(x) = f(a) \).

However, when dealing with discontinuous functions or functions with undefined points, limits can help us understand the behavior of the function near those points. In practice, calculating limits might involve factorization, rationalization, or applying specific limit laws. In the exercise, limits were crucial to assess \( f(x) \) at the boundary point \( x = 2 \) from both left and right, confirming the function's continuity.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. They are like a mathematical patchwork, where each 'patch' has its own rule. These functions are especially useful for modeling situations that change behavior at certain points.

For continuity in piecewise functions, it's essential to consider the points where the function 'switches' from one rule to another. At these points, known as transition points, you'd verify if the function approaches the same value from both sides and if it’s defined at that point. In the function \( f(x) \) from the exercise, the transition occurs at \( x = 2 \) and has been examined to ensure that the pieces of the function 'stitch together' without any gaps or jumps, indicating continuity.

Polynomial Functions

Polynomial functions, which include terms like \( x \), \( x^2 \) , \( x^3 \), and so on, are the basis for understanding more complex functions. They have the general form \( P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \) where \( n \) is a non-negative integer, and the coefficients \( a_0, a_1, ..., a_n \) are real numbers, with \( a_n \) not equal to zero. Polynomials are everywhere continuous, which means they do not have breaks, holes, or jumps in their graphs.

Both component functions in our exercise, \( 3+x \) and \( x^2+1 \) are polynomials, and thus individually continuous everywhere. When dealing with homework involving polynomials, you can confidently say that these parts will not introduce discontinuity, and any potential issues will typically be at the joints in piecewise definitions.