Question

$f\left(x\right)=min\{x,x^2\}\forall x\in R$. Then $f\left(x\right)$ is

• A. discontinuous at 0
• B. discontinuous at 1
• C. continuous on R
• D. continuous at 0, 1

Answer 1

Answer: (C), (D)

continuous on R
continuous at 0, 1

If we look at the function f(x),
We can divide the entire number line into 3 sections.
for x < 0 f(x) = x
for 0 < x < 1 f(x) = $x^2$
for x > 1 f(x) = x
For checking the continuity, the doubtful points are x = 0 and x = 1
At x = 0
lim $x\to 0^{-}f\left(x\right)=x=0$
lim $x\to 0^{+}f\left(x\right)=x^2=0$
At x = 1
lim $x\to 1^{-}f\left(x\right)=x^2=1$
lim $x\to 1^{+}f\left(x\right)=x=1$
Hence, the function is continuous on the entire number line