# How do you use the definition of continuity to determine weather f is continuous at 2-x if x<1, 1 if x=1 and x^2 if x>1?

Feb 9, 2017

see below

#### Explanation:

f(x)={(2-x if x<1) , (1 if x=1), (x^2 if x>1)

Here are the intervals $\left(- \infty , 1\right) \left(1 , \infty\right)$. If we can show that

${\lim}_{x \to {1}^{-}} f \left(x\right) = f \left(1\right)$ and ${\lim}_{x \to {1}^{+}} f \left(x\right) = f \left(1\right)$ then we can prove that f(x) is continuous

${\lim}_{x \to {1}^{-}} f \left(x\right) = {\lim}_{x \to {1}^{-}} 2 - x = 2 - 1 = 1$

$f \left(1\right) = 1$

Since $f \left(1\right) = {\lim}_{x \to {1}^{-}} 2 - x$ f is therefore continuous from the left at 1.

${\lim}_{x \to {1}^{+}} f \left(x\right) = {\lim}_{x \to {1}^{+}} {x}^{2} = {1}^{2} = 1$

$f \left(1\right) = 1$

Since $f \left(1\right) = {\lim}_{x \to {1}^{+}} {x}^{2}$, f is therefore continuous from the right at 1. Hence f is continuous on $\left(- \infty , \infty\right)$