Discuss the continuity of the function `f (x)= |x| " at " x=0 `

Answer

To discuss the continuity of the function \( f(x) = |x| \) at \( x = 0 \), we will follow these steps:

### Step 1: Define the function
The function we are analyzing is:
\[ f(x) = |x| \]

### Step 2: Identify the point of interest
We are interested in the continuity of \( f(x) \) at the point \( x = 0 \).

### Step 3: Check the left-hand limit (LHL)
To find the left-hand limit as \( x \) approaches 0, we consider values of \( x \) that are slightly less than 0 (i.e., \( x \to 0^- \)):
\[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} |x| \]
Since \( x \) is negative in this case, we can express \( |x| \) as:
\[ |x| = -x \quad \text{(for } x < 0\text{)} \]
Thus,
\[ \text{LHL} = \lim_{x \to 0^-} -x = 0 \]

### Step 4: Check the right-hand limit (RHL)
Now, we calculate the right-hand limit as \( x \) approaches 0, considering values of \( x \) that are slightly greater than 0 (i.e., \( x \to 0^+ \)):
\[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} |x| \]
Since \( x \) is positive in this case, we can express \( |x| \) as:
\[ |x| = x \quad \text{(for } x \geq 0\text{)} \]
Thus,
\[ \text{RHL} = \lim_{x \to 0^+} x = 0 \]

### Step 5: Evaluate the function at the point
Next, we find the value of the function at \( x = 0 \):
\[ f(0) = |0| = 0 \]

### Step 6: Check continuity
For \( f(x) \) to be continuous at \( x = 0 \), the following condition must hold:
\[ \text{LHL} = \text{RHL} = f(0) \]
We found:
\[ \text{LHL} = 0, \quad \text{RHL} = 0, \quad f(0) = 0 \]
Since all three values are equal, we conclude that:
\[ f(x) \text{ is continuous at } x = 0. \]

### Final Conclusion
Thus, the function \( f(x) = |x| \) is continuous at \( x = 0 \).

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