Video: Identifying Continuous Functions Graphically

Determine whether the following statement is true or false: The function represented by the graph is a continuous function.

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Video Transcript

Determine whether the following statement is true or false. The function represented by the graph is a continuous function.

For this question, we’ve been given a function 𝑔 of 𝑥, which is defined for all values of 𝑥 over the real numbers, denoted by these arrows. Now, we can almost immediately see that our function 𝑔 of 𝑥 is discontinuous, since at a value of 𝑥 equals three, we have a gap in our graph. In fact, we may recognise this as a jump discontinuity, with 𝑔 of 𝑥 being undefined at the point three, two, denoted by the hollow dot, and defined at the point three, one, denoted by the filled dot.

If we were to look at the left- and right-sided limits, as 𝑥 approaches three, we would find that although both exist, their values disagree. And this would mean that the normal limit does not exist. Here, we recall our definition for continuity at a point, which says that the limit, as 𝑥 approaches 𝑎, of said function must be equal to the value of the function evaluated where 𝑥 is equal to 𝑎. Now, in our case, the limit, as 𝑥 approaches three of 𝑔 of 𝑥, is not equal to 𝑔 of three, since the limit does not exist.

We have, therefore, proved that the function 𝑔 of 𝑥 has a discontinuity at 𝑥 equals three. Our answer to the question is, therefore, false. The function represented by the graph is not a continuous function. As a final point, we may note that our function can have a discontinuity even though it is defined over all real numbers.

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