Working out if a piecewise function is continuous at 0.

Question

I need to prove that the following piecewise function is continuous at 0.

f(x)={ x*3 - 3x, x ≤ 0

{ 1 - Cos x, x>0

I've plotted it on desmos and it sure does look continuous at 0, but how do i formally prove this?

Thank you

Kara x

Answer 1

For this function to be continuous at x=0. the limit of the function must exist and it must equal f(0).

First we have to see if the limits from both sides of 0 are equal.

0³ - 3(0) = 0 The limit from the left of 0 equals 0.

1- cos(0) = 0 The limit from the right of 0 equals 0.

So the limit of the function as x approaches 0 equals 0.

f(0) also equals 0 so this function is continuous at x = 0.

Answer 2

To show that a function is continuous at a certain value you have to show that it is defined at that value and that these values are the same which they are. At x=0 both functions =0 and that their limits are the same which they are