Q.Examine the continuity of the function $f(x) = 2x^2 - 1$ at $x = 3$.

Concept: Continuity At A Point — A function $f$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.

1. Compute $f(3)$:

   $f(3) = 2(3)^2 - 1 = 2(9) - 1 = 18 - 1 = 17$.

2. Compute $\lim_{x \to 3} f(x)$:

   Since $f$ is a polynomial, the limit equals the function value:

   $\lim_{x \to 3} (2x^2 - 1) = 2(3)^2 - 1 = 17$.

3. Compare:

   $\lim_{x \to 3} f(x) = 17 = f(3)$.

The Core Idea: What Does Continuity At A Point Mean?

A function is continuous at a point if three things hold — and they must all be true simultaneously:

1. The function is defined at that point (the value exists).
2. The limit of the function exists as $x$ approaches that point.
3. The limit equals the function’s value.

If any one of these fails, the function is discontinuous there. For a polynomial like $f(x)=2x^2-1$, we expect continuity everywhere — but we still verify it formally, because the reasoning is the same for any function.

Step-by-Step Verification

1. Check that $f(3)$ exists.

Plug $x=3$ directly into the formula:

The function is defined at $x=3$, and its value is $17$.

2. Find the limit of $f(x)$ as $x \to 3$.

Since $f$ is a polynomial, the limit as $x$ approaches any real number is simply the value of the polynomial at that number. This is because polynomials are built from addition, subtraction, and multiplication — operations that behave nicely under limits. Formally:

The limit exists and equals $17$.

3. Compare the limit and the function value.

We have:

They are equal. Therefore, all three conditions for continuity at $x=3$ are satisfied.

Why This Works for Polynomials

A polynomial like $2x^2-1$ is continuous at every real number because it’s built from the constant function and the identity function $x$, both of which are continuous everywhere. The operations of scaling, adding, and multiplying preserve continuity. So the result here is not surprising — but the process of checking is what builds your understanding.

Method: Verifying Continuity at a Point for a Polynomial

When the function is a polynomial (or any function known to be continuous everywhere), verifying continuity at a specific point is not about 'checking if it looks continuous' — it is about writing out the formal three-condition check so the limit and the value are shown to agree explicitly, since that structure is what full-mark board answers require.

Steps

Step 1: Compute the function value at the point

Substitute $x=a$ directly into $f$ to get $f(a)$.

Step 2: Compute the limit as $x\to a$

For a polynomial, direct substitution gives the limit too — the algebraic operations that build a polynomial (addition, multiplication by constants, powers) all preserve limits, so no factoring or cancellation is needed.

Step 3: Compare the limit with the function value

State this equality explicitly — this is the actual continuity condition being verified, not just 'it's a polynomial so it's continuous.'

The general habit: even for the simplest functions, write the limit and the value side by side and show they match — that is what the continuity definition actually asks for.