Question

Prove that the function f(x) = |x| is continuous at x = 0.

Hint:

For proofs of continuity, always follow the three-step structure: calculate \( f(a) \), calculate the LHL and RHL to find \( \lim_{x \to a} f(x) \), and finally state that they are equal. This systematic approach ensures no steps are missed.

Answer 1

Step 1: Understanding the Concept:
A function \( f(x) \) is said to be continuous at a point \( x = a \) if three conditions are met:
\( f(a) \) is defined.
The limit of the function as \( x \) approaches \( a \) exists. This means the Left-Hand Limit (LHL) equals the Right-Hand Limit (RHL). \[ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \] The value of the limit is equal to the value of the function at that point. \[ \lim_{x \to a} f(x) = f(a) \] Step 2: Detailed Explanation:
We need to check these three conditions for the function \( f(x) = |x| \) at the point \( x = 0 \).
Condition 1: Value of the function at x = 0
The value of the function at \( x=0 \) is: \[ f(0) = |0| = 0 \] The function is defined at \( x=0 \).
Condition 2: Existence of the limit at x = 0
We need to calculate the Left-Hand Limit (LHL) and the Right-Hand Limit (RHL).
Left-Hand Limit (LHL): \[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} |x| \] By definition, for \( x<0 \), \( |x| = -x \). So, we can write: \[ \text{LHL} = \lim_{x \to 0^-} (-x) = - (0) = 0 \] Alternatively, using \( h \to 0^+ \): \[ \text{LHL} = \lim_{h \to 0^+} f(0-h) = \lim_{h \to 0^+} |0-h| = \lim_{h \to 0^+} |-h| = \lim_{h \to 0^+} h = 0 \] Right-Hand Limit (RHL): \[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} |x| \] By definition, for \( x>0 \), \( |x| = x \). So, we can write: \[ \text{RHL} = \lim_{x \to 0^+} (x) = 0 \] Alternatively, using \( h \to 0^+ \): \[ \text{RHL} = \lim_{h \to 0^+} f(0+h) = \lim_{h \to 0^+} |0+h| = \lim_{h \to 0^+} |h| = \lim_{h \to 0^+} h = 0 \] Since LHL = RHL = 0, the limit exists and \( \lim_{x \to 0} f(x) = 0 \).
Condition 3: Comparing the limit and function value
We have found that: \[ \lim_{x \to 0} f(x) = 0 \text{and} f(0) = 0 \] Therefore, \( \lim_{x \to 0} f(x) = f(0) \).
Step 3: Final Answer:
Since all three conditions for continuity are satisfied, the function \( f(x) = |x| \) is continuous at \( x = 0 \).