Determine for what numbers, if any, the function is discontinuous. Construct a table to find any

Determine for what numbers, if any, the function is...

Key Concepts

Continuity

Continuity of a function at a point means that the function is defined at that point, its limit exists as the variable approaches the point, and the limit value is equal to the function’s value. Understanding continuity is key to analyzing whether a function behaves predictably and without breaks at specific points.

Discontinuity

Discontinuity occurs when one or more of the conditions for continuity is not met at a particular point. This can be classified into types like removable, jump, or infinite discontinuities. Recognizing the type of discontinuity helps in determining if it can be 'fixed' by redefining the function at that point or if it is an inherent feature of the function.

Limits

Limits describe the behavior of a function as its input approaches a certain point. They are fundamental in understanding and defining continuity, especially around points where the function's definition might change. Limits are used to formally assess whether a function approaches a specific value from different directions.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. Evaluating continuity for piecewise functions requires checking that the various pieces connect smoothly at the boundaries where the definition changes, often using limits from both sides of these points.

Removable Discontinuity

A removable discontinuity occurs at a point where a function is not defined or does not match the limit, but the limit exists. This discontinuity can 'potentially' be removed by redefining the function at that point, making it continuous, which is a critical concept in function analysis.

Table of Values for Limits

Using a table of values is a practical method to approximate the limit of a function at a point by computing the function’s values as the independent variable gets arbitrarily close from both sides. This approach helps in visualizing and verifying the behavior of the function near potential discontinuities.

Transcript

00:02 So we have the following function for f of x, and we need to prove that whether it's continuous for all values.

00:09 But if not, well, what are the values for which is not continuous? so let's look at the fraction that we have.

00:19 Cosine of x over x minus pi over 2.

00:23 Both are polynomial.

00:24 Well, not polynomials, but cosine of x is continuous for all values.

00:30 And x minus pi over 2 is a polynomial, which is continuous for.

00:34 Values.

00:37 So therefore we don't have anything to worry about except when the denominator is zero and that happens as side of it still made.

00:46 So therefore for the fraction, we know it's always continuous, which means you have only one value left to check and that is pi over two because that's the breaking point between the function and when it's equal to one.

01:02 Okay, so for which let's not check how to prove if it's continuous at pi halves.

01:13 Well, for that we need to prove that f of pi half is equal to the limit as x approaches pi half from the left of f of x, and that is equal to the limit as x approaches pi half from the right side of f of x.

01:46 All right, so what is f of pi half? well, we have it here when x is equal to pi half is going to be one.

01:58 So f of pi half is going to be one.

02:01 Next, let's look at our limit.

02:03 So if x approaches pi half from the left and the right, that means x is not equal to pi half...

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