Prove that the functions f(x)=x+2, if x=1 are discontinuous.

We notice that the point where we have to verify the
continuity of the function is x = 1.

The function is
continuous over the ranges (-infinite ; 1) and (1 ;
+infinite).

Now, we'll verify the continuity of the
function by evaluating the lateral limits of the
function.

For x<1, we'll calculate the limit of the
function f(x) = x+2.

lim f(x) = lim (x+2) = 1 + 2 =
3

For x>1, we'll calculate the limit of the function
f(x) = x^2.

lim f(x) = lim x^2 = 1^2 =
1

The values of the lateral limits are finite
but they are different, so the function is discontinuous in the points x =
1.