(i) Dissusse the continuity of the function `f(x)={(|x-a|", " xne a ),(" "0 ", "x=a):}` at `x=a` (ii) Discuss the continutiy of the function `f(x)={(|x-3|/(x-3)", " xne 3 ),(" "0 ", "x=3):}` at `x=3`

Answer

To discuss the continuity of the given functions, we will follow the standard definition of continuity at a point. A function \( f(x) \) is continuous at \( x = c \) if:

1. \( f(c) \) is defined.
2. The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
3. The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

### Part (i): Continuity of the function \( f(x) = \begin{cases} |x-a| & \text{if } x \neq a \\ 0 & \text{if } x = a \end{cases} \) at \( x = a \)

**Step 1: Check if \( f(a) \) is defined.**
- We have \( f(a) = 0 \).

**Step 2: Calculate the left-hand limit as \( x \) approaches \( a \).**
- For \( x < a \), \( f(x) = |x-a| = -(x-a) = a - x \).
- Thus,
\[ \lim_{x \to a^-} f(x) = \lim_{x \to a^-} (a - x) = a - a = 0. \]

**Step 3: Calculate the right-hand limit as \( x \) approaches \( a \).**
- For \( x > a \), \( f(x) = |x-a| = x - a \).
- Thus,
\[ \lim_{x \to a^+} f(x) = \lim_{x \to a^+} (x - a) = a - a = 0. \]

**Step 4: Compare the limits and the function value.**
- We have:
\[ \lim_{x \to a^-} f(x) = 0, \quad \lim_{x \to a^+} f(x) = 0, \quad f(a) = 0. \]
- Since both limits are equal and equal to \( f(a) \), the function is continuous at \( x = a \).

### Conclusion for Part (i):
The function \( f(x) \) is continuous at \( x = a \).

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### Part (ii): Continuity of the function \( f(x) = \begin{cases} \frac{|x-3|}{x-3} & \text{if } x \neq 3 \\ 0 & \text{if } x = 3 \end{cases} \) at \( x = 3 \)

**Step 1: Check if \( f(3) \) is defined.**
- We have \( f(3) = 0 \).

**Step 2: Calculate the left-hand limit as \( x \) approaches \( 3 \).**
- For \( x < 3 \), \( f(x) = \frac{|x-3|}{x-3} = \frac{-(x-3)}{x-3} = -1 \).
- Thus,
\[ \lim_{x \to 3^-} f(x) = -1. \]

**Step 3: Calculate the right-hand limit as \( x \) approaches \( 3 \).**
- For \( x > 3 \), \( f(x) = \frac{|x-3|}{x-3} = \frac{x-3}{x-3} = 1 \).
- Thus,
\[ \lim_{x \to 3^+} f(x) = 1. \]

**Step 4: Compare the limits and the function value.**
- We have:
\[ \lim_{x \to 3^-} f(x) = -1, \quad \lim_{x \to 3^+} f(x) = 1, \quad f(3) = 0. \]
- Since the left-hand limit and right-hand limit are not equal, and neither is equal to \( f(3) \), the function is not continuous at \( x = 3 \).

### Conclusion for Part (ii):
The function \( f(x) \) is not continuous at \( x = 3 \).

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