have / having a cold

[Tan Elaine]

I have been taught that we have to say "I have a cold", not "I am having a cold."

Despite the fact that he had a cold, he went to work. (I know this sentence is fine.)

But I wonder if the following sentence is correct? If it is, why is 'having' correct in the sentence, but not in the above underlined sentence.

Despite having a cold, he went to work.

Thanks.

[Larkus]

I'm not a teacher.

"I am having a cold."

Despite having a cold, he went to work.

The second sentence is correct. The difference between the two verb forms is that the former is the first person present continuous, the latter is the gerund of "to have".