Question

determine the intervals of the domain over which the function is continuous.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is continuous on box.
(type your answer in interval notation.)
b. the function is not continuous.

Explanation:

🆕 New Concept Discovered: Interval Continuity
Tracing a graph without lifting your pencil.

Step 1: Understand continuity on a graph

A function is continuous on an interval if you can trace its graph over that interval from left to right without lifting your pencil. This means there are no holes, breaks, jumps, or vertical asymptotes in the graph.

Step 2: Analyze the given graph

Looking at the provided graph:

• On the left side, the blue curve has an arrow pointing downwards and to the left, starting from \(x = 1\). This indicates the graph continues infinitely to the left toward \(-\infty\).
• As we move from left to right, the curve goes smoothly through the points without any breaks or gaps.
• It passes through \(x = 4\), stays flat along the x-axis between \(x = 5\) and \(x = 6\), and then curves upwards.
• On the right side, the curve has an arrow pointing upwards and to the right, indicating it continues infinitely to the right toward \(\infty\).

Since there are no breaks, holes, or jumps anywhere along the entire curve, the function is continuous over its entire domain.

Step 3: Write the domain in interval notation

Since the graph extends infinitely in both directions without any interruption, the interval of continuity is all real numbers.

In interval notation, this is written as:

$$ (-\infty, \infty) $$

Answer:

A. The function is continuous on \((-\infty, \infty)\).