Question

a. Sketch the graph of a function that is not continuous at \(1,\) but is defined at 1. b. Sketch the graph of a function that is not continuous at \(1,\) but has a limit at 1.

Short answer

Question: Sketch a function that is (a) not continuous at 1, but defined at 1, and (b) not continuous at 1, but has a limit at 1. Answer: a) One example of such a function can be the piecewise function: $$ f(x) = \begin{cases} x & \text{if}\ x<1 \\ 3 & \text{if}\ x=1 \\ x+2 & \text{if}\ x>1 \end{cases} $$ This function has a jump at x=1 with the point (1,3). b) Another example of such a function is: $$ g(x) = \frac{x^2-1}{x-1} $$ This function simplifies to g(x)=x+1 when x≠1, and is not defined at x=1. The graph of this function has a hole at x=1 and the limit as x approaches 1 is 2.

Step-by-step solution

Identify a function with a hole or jump at x=1

A function that is not continuous at \(1\) but defined at \(1\) could have a hole or a jump. One type of such function is a piecewise function. Let's choose the following piecewise function, which has a jump at \(x=1\): $$ f(x) = \begin{cases} x & \text{if}\ x<1 \\ 3 & \text{if}\ x=1 \\ x+2 & \text{if}\ x>1 \end{cases} $$

Sketch the graph

First, plot \(f(x)=x\) for \(x<1\). This portion of the graph will be a straight line with a slope of \(1\) that stops at \(x=1\), when \(y=1\). Then, plot the single point \((1,3)\). This is the point where the function is defined but not continuous. There will be a "jump" in the graph at this point. Lastly, plot \(f(x)=x+2\) for \(x>1\). This portion of the graph will be a straight line with a slope of \(1\) that starts at \(x=1\), when \(y=3\). The graph of the function should have a jump at \(x=1\) with the point \((1,3)\). #b. Part (b): Sketch a function not continuous at 1, but has a limit at 1.

Identify a function that approaches a limit at x=1

A function that is not continuous at \(1\) but has a limit at \(1\) could have a hole in its graph. Let's choose the following function with a hole at \(x=1\): $$ g(x) = \frac{x^2-1}{x-1} $$ This function simplifies to \(g(x)=x+1\) when \(x\neq1\). When \(x=1\), the function is not defined because of division by zero. The graph of the simplified function will approach a specific value at \(x=1\) but will not be defined at that point.

Sketch the graph

First, plot \(g(x)=x+1\) for all values of \(x\neq1\). This portion of the graph will be a straight line with a slope of \(1\) that passes through the origin. There will be a hole in the graph at \(x=1\), when \(y=1+1=2\). Next, observe that the function is not defined at \(x=1\). As \(x\) approaches \(1\), the graph will approach the value \(y=2\), but there will be a hole in the graph at that point. The graph of the function should have a hole at \(x=1\) and the limit as \(x\) approaches \(1\) is \(2\).

Key concepts

Continuity

Continuity in a function refers to the smooth, unbroken pathway of its graph. For a function to be continuous at a point, three key conditions must be satisfied: the function must be defined at the point, the limit of the function as it approaches that point must exist, and the value of the function at that point must equal the limit. If all these conditions are met, the function does not experience any jumps, holes, or breaks at that point.

For example, consider a basic linear function like \( y = x \), which is continuous for all values of \( x \). This means that as \( x \) moves from one point to another, the curve is seamless without any interruptions. Since the value of the function at any point equals the limit of the function as it approaches that point, continuity is achieved.

Discontinuity

Discontinuity occurs when a function breaks or skips a point on its curve. This can happen for several reasons, such as jumps, holes, or asymptotes. A simple example is a piecewise function, which is made up of different expressions for different intervals of \( x \).

• A **jump discontinuity** happens when there's an immediate change in the function's value at a specific point. Using the piecewise function from the original exercise, \( f(x) = \begin{cases} x & \text{if} \ x<1 \ 3 & \text{if} \ x=1 \ x+2 & \text{if} \ x>1 \end{cases} \), there is a jump at \( x = 1 \).
• A **hole discontinuity** occurs when the function approaches a specific value but is not actually defined at that point. Consider \( g(x) = \frac{x^2-1}{x-1} \), which creates a hole at \( x = 1 \).

Identifying discontinuities involves examining where the function might not adhere to standard "continuous" behavior, such as in the case of a breaking line or an undefined point.

Limits

The concept of limits helps us understand the behavior of functions as they approach a certain point. Limits determine the value that a function's output approaches as the input gets closer to a specific point, even if the function isn’t defined at that point.

When examining the function \( g(x) = \frac{x^2 - 1}{x - 1} \), it simplifies to \( g(x) = x + 1 \) for all \( x eq 1 \). Despite the function not being defined at \( x = 1 \), the limit as \( x \) approaches 1 is \( x + 1 = 2 \). This demonstrates that the curve tends toward \( y = 2 \) as it nears \( x = 1 \).

The importance of limits comes into play particularly in calculus and analysis where dealing with continuous growth or change is vital, even through points that aren't precisely defined but still pivotal in understanding overall function behavior.