Q.Discuss the continuity of the function $f$ given by $f(x) = \begin{cases} x, & \text{if } x \geq 0 \\ x^2, & \text{if } x < 0 \end{cases}$.

The only possible break is at $x=0$, where the rule changes.

• Left limit: $\lim_{x\to 0^-}x^2 = 0$
• Right limit: $\lim_{x\to 0^+}x = 0$
• Value: $f(0) = 0$ (since $0\ge 0$ uses $f(x)=x$) …

Where to look

Continuity at $x=a$ needs $f(a)$ defined, $\lim_{x\to a}f(x)$ to exist, and the two to be equal. The pieces $x$ and $x^2$ are polynomials, continuous on their own, so only the join at $x=0$ needs checking.

Value at the point

Since $0\ge 0$, the top rule applies: $f(0)=0$.

One-sided limits

From the left ($x<0$, $f(x)=x^2$):

From the right ($x>0$, $f(x)=x$):

Compare

Both one-sided limits are $0$, so $\lim_{x\to 0}f(x)=0$, and this equals $f(0)=0$. All three continuity conditions hold at $x=0$.

Method: Confirming Continuity Across a Piecewise Boundary (When It Actually Holds)

This method applies to the same class of piecewise-boundary problems as above, but covers the case where the three conditions all end up matching — i.e., the function turns out to be continuous everywhere, including at the boundary.

Steps

Step 1: Locate the only point that needs checking.

As always, every piece away from the boundary is continuous on its own (here, $x$ and $x^2$ are both polynomials), so only the single junction point needs the full three-condition test.

Step 2: Evaluate $f(a)$ using the piece that the definition explicitly assigns to $x=a$.

Piecewise definitions often use $\geq$ or $\leq$ (rather than strict $<,>$) on exactly one side to pin down which formula governs the boundary point itself — read this carefully.

Step 3: Compute both one-sided limits. …

Common Mistakes

Mistake 1: Assuming a "different formula on each side" automatically means a jump/discontinuity.

Why it's wrong: two different-looking expressions can still meet at exactly the same value at the boundary, giving a smooth join — here $x$ and $x^2$ both equal $0$ at $x=0$, so there is no break at all. Correct approach: never assume a discontinuity from the shape of the definition alone; always compute the actual one-sided limits and compare.

Mistake 2: Using the wrong piece to evaluate $f(0)$ itself. …