Can a discontinuous function on a closed interval be integrable on that interval?

There is a theorem of the calculus that you can look up in Larson's book Volume I, theorem 4.4 and it is on page 307 that goes like this:

Theorem 4.4: Continuity implies integrability

If a function f is continuous in the closed interval [a , b], then f is integrable in [a , b].

Clearly this theorem is answering the question I am asking in the title of the post, that is to say that for f to be integrable f has to be continuous, therefore a discontinuous function cannot be integrable in the interval where it presents the discontinuity.

Apart from what I have just explained with Theorem 4.4, there is also the reciprocal of this theorem, i.e. if f is integrable in a closed interval [a , b], then f is continuous in the closed interval [a , b].

To prove Theorem 4.4 and its reciprocal I want to propose the following:

Example

Determine if the function:is integrable in closed interval [3 ; 5] Explain the answer.

It is clearly seen that the function f(x) = 1 / x-4 is discontinuous for x = 4, since the zero denominator is made, which implies that there is a discontinuity in the closed interval [3 ; 5], which we can corroborate with the graph of the function f(x) = 1 / x-4 in geogebra software to evaluate its continuity:

We can see in the previous graph that the blue line x = 4 is a vertical asymptote, which certifies that in the closed interval [3 ; 5] there is a discontinuity at the point x = 4, therefore in this interval [3 ; 5] the function f(x) = 1 / x-4 function is not integrable, which we will demonstrate below:

We could see that when we evaluate the result of the indefinite integral of Ln(x-4) in the closed interval [3;5] we get a discontinuity, which implies that the function is not integrable in the closed interval [3;5].

To prove the reciprocity of Theorem 4.4, we must simply analyze that a function that is not integrable on a closed interval [a;b] makes the function discontinuous and even non-integrable on the closed interval [a;b].

Recommended Bibliographic Reference

Larson and Hosttler's Book of Calculus with Analytic Geometry. Volume I. 7th edition.

Note: The graph of the rational function was generated by geogebra software and edited in Microsoft PowerPoint. The images of the equations were elaborated using the equation insertion tools of Microsoft PowerPoint.