Just a question about continuity of functions

[o_O]

Just a bit curious about something. I know that if two functions f(x) and g(x) are continuous, then so would f(x) + g(x) and f(x)g(x).

However, is that the case for discontinuous functions? i.e. If f(x) and g(x) were both individually not continuous functions, does it follow that f(x) + g(x) and f(x)g(x) aren't continuous as well? Something tells me that they aren't but I just can't figure what.

 

[stapel]

I think the answer would depend upon the functions. The following are discontinuous, each having a "break" at x = 0:

. . . . .\(\displaystyle \L f(x)\, =\, \left{\begin{array}{rr}-1\,&\mbox{for }x\, <\, 0\\1\,&\mbox{for }x\,\geq\,0\end{array}\)

. . . . .\(\displaystyle \L g(x)\, =\, \left{\begin{array}{rr}1\,&\mbox{for }x\, <\, 0\\-1\,&\mbox{for }x\,\geq\,0\end{array}\)

But the sum, h(x) = f(x) + g(x), is clearly continuous, since it is the constant function, h(x) = 0.

Eliz.

 

[o_O]

Ooh that's smart. Thanks a lot!