Q.Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} 2x+3, & \text{if } x \leq 2 \\ 2x-3, & \text{if } x > 2 \end{cases}$

The key idea is continuity at a point: a function is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$. For a piecewise function, we check the left-hand limit, right-hand limit, and the function value at the boundary.

Step 1: At $x = 2$, the function value is $f(2) = 2(2) + 3 = 7$.

Step 2: Left-hand limit: $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (2x+3) = 2(2)+3 = 7$. …

The Core Idea: Continuity at a Point

A function is continuous at a point $x = a$ if three things happen together:

1. The function is defined at $a$ — $f(a)$ exists.
2. The limit of $f(x)$ as $x$ approaches $a$ exists (both sides agree).
3. That limit equals $f(a)$.

For a piecewise function, the only place where things can go wrong is at the "seam" — the point where the definition changes. Here, that seam is $x = 2$. Everywhere else, each piece is a straight line, and straight lines are continuous everywhere. So the entire question boils down to: what happens at $x = 2$?

Step-by-Step Analysis

1. Check the function value at $x = 2$

Since $x = 2$ falls in the first case ($x \leq 2$), we use $f(x) = 2x + 3$.

So $f(2) = 7$. That's fine — the function is defined.

2. Compute the left-hand limit as $x \to 2^-$

When $x$ approaches $2$ from the left (values like $1.9, 1.99, 1.999$), we are still in the region $x \leq 2$, so we use the same piece: $2x + 3$.

The left-hand limit is $7$, which matches $f(2)$.

3. Compute the right-hand limit as $x \to 2^+$

When $x$ approaches $2$ from the right (values like $2.1, 2.01, 2.001$), we are in the region $x > 2$, so we use the other piece: $2x - 3$.

The right-hand limit is $1$.

4. Compare the two one-sided limits

Left-hand limit: $7$

Right-hand limit: $1$

Since $7 \neq 1$, the two-sided limit $\displaystyle \lim_{x \to 2} f(x)$ does not exist.

5. Apply the continuity condition …

Method: Finding All Points of Discontinuity of a Piecewise Function

For a piecewise function built from continuous pieces (polynomials, linear expressions, absolute values), discontinuities can only ever occur at the boundary point(s) where the definition switches from one rule to another — never inside a single piece.

Steps

Step 1: Identify every boundary/seam value

List every $x$-value where the piecewise definition changes rule (here, $x=2$).

Step 2: Note that each individual piece is continuous on its own open interval

Since each piece (e.g. $2x+3$, $2x-3$) is itself a polynomial, it is automatically continuous everywhere it's used — so no work is needed away from the seam(s).

Step 3: At each seam, compare the function value with both one-sided limits

using whichever piece's formula applies on each side.

Step 4: Classify the seam based on the comparison …

Common Mistakes

Mistake 1: Assuming that because both pieces are the same type (both linear), the function must be continuous

Why it's wrong: the type of formula (linear, polynomial) guarantees continuity only within its own piece, not across the seam — two lines can easily have different values at the meeting point even if they share a slope. Correct approach: always compute the actual one-sided limits and function value numerically at the seam rather than reasoning from the shape of the pieces.

Mistake 2: Checking continuity at points other than the seam …