Help. Whats the domain of this function? is it continuous

The question was in my calculus book.

Let f(x)= -1/x and g(x)=(1/x)

They have an infinite discontinuity at x=0 clearly. domain for both functions is x not equal to 0

if f(x) and g(x) are both discontinuous at x=c, is f(x)+g(x) continuous at x=c?

this book is vague, but it gave me f(x) and g(x) to consider

This seems contradicting given that

continuity is lim x->c f(x)=f(c)

the domain of f(x)+g(x) is x not equal is zero

the premise is that a value cant be divided by zero..

Although that these functions add to equal zero. we cant let x equal zero, unless we redefine it to be 0.

My questions,

1) why arent they reffering that f(x)+g(x) is continuous on its domain?

because f(x)+g(x) is the line y=0 with a removable discontinuity

2) Does removeable discontinuity apply continuity??


If you know of any deeper definitions of continuity than the one listed above. Please list it

I think you've already answered your question.
"f(x)+g(x) is the line y=0 with a removable discontinuity"
As the name indicates, a discontinuity is a place where the function is not continuous.

h(x) = -1/x + 1/x
h(x) = 0 for any value of x except x = 0, where h(x) is undefined.

The limit definition is THE definition of continuity. Since h(0) is undefined, it is not true that lim x-> 0 h(x) = h(0). The function is not continuous.

You have the right answer (I think. Your post was a bit hard to follow.). Both f(x) and g(x) don't exist at x=0, so they have no value at that point, and of course you can't add things that don't exist.

Continuity (as I learned it) is just that there are no discontinuities. In this case, I wouldn't call f(x)+g(x) continuous, but I know some people who would.

It really depends on the definition of c in your definition of continuity. If c is ANY x, even ones not in the domain, then it's not continuous. If c is any x in the domain, then it is.

Sorry for rambling a bit, though. : l