Prove that f(x)=x is continuous

Answer 1

See Patrick's answer for the epsilon delta limit method of proof of continuity.

But the function f(x) = x may have discontinuities if the domain is not all real numbers.

IF the domain is just integers or natural numbers, then f(x) is everywhere discontinuous. It has endless discontinuous points. More holes than Swiss cheese.

Answer 2

Proves:

limit f(x) as x-->c is f(c)=c

Definition:

for every E>0 there exists d(E)>0 such that

f is in neighborhood of c whenever x in neighborhood of c

where E is the "epsilon" and d is the delta in terms

of epsilon per the definition.

Then

C-e< f(x)=x < C+e whenever C-d < x <C+d

suggests to let D=E>0

So

then when C-d < x < C+d

C-E < x < C-E<-- replaces d with d=E

C-E < f(x) < C-E repalaces x with f(x)=x

so the function is in some neighborhood of c

when x is in some neighborhood of c.

Therefore the limit of f(x)=c as x-->c is c, which

agrees with the function value f(c)=c

This satisifies the definiton of continuity.

Loosely, f(x)=x is a polynomial (in fact linear) function

which is continuous