Question

Tell whether if the following piecewise function is a continuous at a given point or not. (SHOW THE SOLUTION).

1. at x = 4

x + 1 if x < 4

(x - 4)² + 3 if x ≥ 4

Answer 1

1.

$$\lim\limits_{x\to4^-}f(x)=\lim\limits_{x\to4^-}(x+1)=4+1=5$$ $$\lim\limits_{x\to4^+}f(x)=\lim\limits_{x\to4^+}((x-4)^2+3)=(4-4)^2+3=3$$ $$\lim\limits_{x\to4^-}f(x)=5\not=3=\lim\limits_{x\to4^+}f(x)$$ $$\lim\limits_{x\to4}f(x)=\text{does not exist}$$

The function $f(x)$ is not continuous at $x=4.$

The function $f(x)$ has a jump discontinuity at $x=4.$