Question

If f(x) is discontinuous, determine the reason. f(x)= {x^2 +4, x<=1; x+3, x>1}

Answer 1

Answer:

Observe that f(x) is a continuous function when $x1$ because is a polynomial. The possible problem may occur in x=1.

Then, f(x) is discontinuous in x=1 if the limits of f to the right and the left of 1 exist and are different or if some of those limits doesn't exist.

Let's calculate the limits:

$lim_{x\rightarrow 1^+}f(x)=lim_{x\rightarrow 1}(x+3)=1+3=4$

$lim_{x\rightarrow 1^-}f(x)=lim_{x\rightarrow 1^-}(x^2+4)=1^2+4=5$

Since, $lim_{x\rightarrow 1^-}\neq lim_{x\rightarrow 1^+}$ then f(x) is discontinuous in x=1.