Continuity

Front

What is continuity in mathematics?

Back

Continuity refers to a property of a function where it is uninterrupted and has no breaks, jumps, or holes in its graph.

Front

What does it mean for a function to be continuous at a point x = a?

Back

A function is continuous at x = a if the following three conditions are met: 1) f(a) is defined, 2) the limit of f(x) as x approaches a exists, and 3) the limit equals f(a).

Front

What is a polynomial function?

Back

A polynomial function is a function that can be expressed in the form f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

Front

Is a polynomial function continuous everywhere?

Back

TRUE. Polynomial functions are continuous for all real numbers.

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What is removable discontinuity?

Back

Removable discontinuity occurs at a point where a function is not defined, but can be made continuous by defining or redefining the function at that point.

Front

What is infinite discontinuity?

Back

Infinite discontinuity occurs when the function approaches infinity or negative infinity as it approaches a certain point.

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What is jump discontinuity?

Back

Jump discontinuity occurs when the left-hand limit and right-hand limit at a point exist but are not equal, causing a 'jump' in the graph.

Front

How can you determine if a function is continuous at a specific point?

Back

To determine continuity at a point, check if the function is defined at that point, if the limit exists, and if the limit equals the function's value at that point.

Front

What is the limit of a function?

Back

The limit of a function is the value that the function approaches as the input approaches a certain point.

Front

What does it mean if a function is discontinuous at x = 0?

Back

It means that at x = 0, the function either does not exist, has a jump, or approaches infinity.