Next, we calculate the right-hand limit as x approaches 0. Since x is approaching 0 from the right side, x ≥ 0, so we use the definition f(x) = x. Substituting x=0 into x gives us 0.

Discuss the continuity of the function $f$ given by $f(x)=|x|$ at $x=0$ .

Answer: The function $f(x)=|x|$ is continuous at $x=0$ .

Step-by-step solution

Step 1: Define the absolute value function

The absolute value function, $f(x) = |x|$ , can be defined piecewise. For non-negative values of $x$ , $f(x)$ is simply $x$ . For negative values of $x$ , $f(x)$ is $-x$ , which makes the output positive.

Step 2: Check the function value at $x=0$

To check for continuity at $x=0$ , we first need to find the value of the function at this point. Substituting $x=0$ into the function, we get $f(0) = |0|$ , which is $0$ .

Step 3: Calculate the left-hand limit

Now, we calculate the left-hand limit as $x$ approaches $0$ . Since $x$ is approaching $0$ from the left side, $x < 0$ , so we use the definition $f(x) = -x$ . Substituting $x=0$ into $-x$ gives us $0$ .

Step 4: Calculate the right-hand limit

Next, we calculate the right-hand limit as $x$ approaches $0$ . Since $x$ is approaching $0$ from the right side, $x \ge 0$ , so we use the definition $f(x) = x$ . Substituting $x=0$ into $x$ gives us $0$ .

Step 5: Compare limits and function value

For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the function value at that point must all be equal. In this case, all three values are $0$ .

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Discuss the continuity of the function fff given by f(x)=∣x∣f(x)=|x|f(x)=∣x∣ at x=0x=0x=0.