I believe your econometrics professor was thinking something along the following lines.
Consider the function $F$ with domiain $[0, 1]$ defined by
$$F(x) = \frac{1}{2}x \ \text{for} \ x < \frac{1}{2} $$
$$F(x) = \frac{1}{2}x + \frac{1}{2} \ \text{for} \ x \geq \frac{1}{2} $$
This is a discontinuous function, but a completely valid CDF for some probability distribution on $[0, 1]$. Note that, using this distribution
$$ P\left(\left\{\frac{1}{2}\right\}\right) = \frac{1}{2} $$
There is no function $f$ that serves as a PDF for this distribution, even though there is a CDF.
It's easy enough to check this is true in this simple example if you've seen this kind of thing before. Suppose there is such a pdf $f$, we will show it must have an impossible property, and hence cannot exist.
By the definition of a PDF, we must have
$$ \int_0^x f(t) dt = F(x) - F(0) = \frac{1}{4}x $$
for all $0 < x < \frac{1}{2}$. A function that integrates to a linear function must be constant (technically constant almost everywhere), so we conclude that
$$ f(x) = \frac{1}{4} \ \text{for} \ x < \frac{1}{2} $$
In the same way, but integrating starting at one, moving towards zero, and ending at $x > \frac{1}{2}$, we reach the same conclusion
$$ f(x) = \frac{1}{4} \ \text{for} \ x > \frac{1}{2} $$
So we have determined $f$ everywhere except for $f\left(\frac{1}{2}\right)$. But it really does not matter what $f\left(\frac{1}{2}\right)$ is, it cannot have the integration property desired. Since
$$ P\left(\left\{\frac{1}{2}\right\}\right) = \frac{1}{2} $$
we would need
$$ \int_{\frac{1}{2} - \epsilon}^{\frac{1}{2} + \epsilon} f(t) dt > \frac{1}{2} $$
for every interval containing $\frac{1}{2}$. But in fact the value of any integral is unaffected by changing the value of a function at any single point, so
$$ \int_{\frac{1}{2} - \epsilon}^{\frac{1}{2} + \epsilon} f(t) dt = \int_{\frac{1}{2} - \epsilon}^{\frac{1}{2} + \epsilon} \frac{1}{4} dt = \frac{1}{2} \epsilon $$
So there's no way out, a function such as $f$ cannot exist.
You can recover the spirit of a PDF, but you must use more sophisticated mathematical objects, either a measure or a distribution.