Question

How do you use the definition of continuity to determine weather f is continuous at `f(x)= x-4` if `x<=0` and `x^2+x-4` if x>0?

The way in which the function `f` is defined, it seems that its continuity is to be discussed at the pt. `x=0`.

Answer 1

`f` is continuous at `x=0`, as discussed in the Explanation Section below.

Explanation:

The function `f` is continuous at `x=0 iff lim_(xrarr0) f(x)=f(0)`.

As `xrarr0` from RHS, i.e., `x rarr0+, x>0, so, f(x)=x^2+x-4`, a quadratic poly., known to be cont. on `RR`

`:. lim_(xrarr0+) f(x)=lim_(xrarr0+) x^2+x-4=-4....(1)`.

As `xrarr0-, x<0, so, f(x)=x-4,` a linear poly., known to be cont. on `RR`.

`:. lim_(xrarr0-) f(x)=lim_(xrarr0-) x-4=-4..............(2)`.

Also, `f(0)=[x-4]_(x=0)=-4................(3)`

From `(1)-(3), lim_(xrarr0) f(x)=f(0)`.

Hence, `f` is continuous at `x=0`