Continuity and Differentiability

Front

What is continuity in a function?

Back

A function is continuous at a point x = c if the following three conditions are met: 1) f(c) exists, 2) lim (x -> c) f(x) exists, and 3) lim (x -> c) f(x) = f(c).

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What is differentiability in a function?

Back

A function is differentiable at a point if it has a defined derivative at that point, meaning the function is smooth and has no sharp corners or discontinuities.

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What does it mean for a function to be continuous but not differentiable?

Back

A function can be continuous at a point but not differentiable if it has a sharp corner or cusp at that point.

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What is removable discontinuity?

Back

A removable discontinuity occurs when a function is not defined at a point, but can be made continuous by defining it appropriately at that point.

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What is infinite discontinuity?

Back

Infinite discontinuity occurs when the function approaches infinity as it approaches a certain point from either side.

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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

What is the relationship between continuity and differentiability?

Back

If a function is differentiable at a point, it must be continuous at that point. However, a continuous function is not necessarily differentiable.

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How can you determine if a function is not differentiable at a point?

Back

A function is not differentiable at a point if it has a sharp corner, cusp, vertical tangent, or if it is discontinuous at that point.

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What is the significance of the limit in determining continuity?

Back

The limit of a function as x approaches a point must equal the function's value at that point for the function to be continuous.

Front

What is a critical point in the context of differentiability?

Back

A critical point is where the derivative of a function is zero or undefined, which may indicate a local maximum, minimum, or point of inflection.