Answer:
Continuity of an equation refers to the absence of abrupt changes or discontinuities in the values of the function, ensuring that the function is defined, its limit exists and is finite, and the value of the function matches the limit at each point within its domain.
Explanation:
Continuity is a fundamental concept in mathematics that describes the behavior of a function or equation. Specifically, continuity refers to the absence of any abrupt changes, breaks, or discontinuities in the values of a function over its domain.
For an equation to be continuous, it must meet the following conditions:
1. The function must be defined at every point within its domain. This means that there are no values of the independent variable (usually denoted as "x") for which the function is undefined.
2. The limit of the function as the independent variable approaches a particular point within its domain must exist and be finite. In other words, the function must approach a specific value as x gets arbitrarily close to a given point.
3. The value of the function at the specific point mentioned above must be equal to the limit mentioned in the previous condition. This ensures that there are no abrupt changes or jumps in the function's values at that point.
In simpler terms, a function is continuous if you can draw its graph without lifting your pen from the paper. There are no holes, jumps, or vertical asymptotes in the graph.
If an equation fails to meet any of the above conditions, it is considered discontinuous. Discontinuities can occur for various reasons, such as having a removable or non-removable discontinuity, a jump, or an essential singularity.