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.
