Complete the following sentences in terms of a limit.


b. A function is continuous from the right at a if _____ .

Verified step by step guidance

1. To understand continuity from the right at a point 'a', we need to consider the behavior of the function as it approaches 'a' from the right side.
2. The formal definition involves limits. Specifically, a function f(x) is continuous from the right at a point 'a' if the right-hand limit of f(x) as x approaches 'a' is equal to the function value at 'a'.
3. In mathematical terms, this can be expressed as: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>lim</mo><mrow><msub><mi>x</mi><mo>&#x2192;</mo></msub><msup><mi>a</mi><mo>+</mo></msup></mrow><mo>f</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>f</mo><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math>
4. This means that as x gets arbitrarily close to 'a' from values greater than 'a', the function values f(x) should approach f(a).
5. If this condition is satisfied, we say that the function is continuous from the right at the point 'a'.

Key Concepts

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. It is a fundamental concept in calculus that helps in understanding the behavior of functions near specific points, especially when they are not explicitly defined at those points.

Continuity

A function is continuous at a point if there is no interruption in its graph at that point. This means that the limit of the function as it approaches the point from both sides equals the function's value at that point. Continuity is essential for ensuring that small changes in input result in small changes in output.

Right-Hand Limit

The right-hand limit of a function at a point 'a' refers to the value that the function approaches as the input approaches 'a' from the right (values greater than 'a'). For a function to be continuous from the right at 'a', the right-hand limit must equal the function's value at 'a'.