Q.Prove that the function $f(x) = 5x - 3$ is continuous at $x = 0$, at $x = -3$ and at $x = 5$.

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

Step 1: For any $a$, $f(a) = 5a - 3$.

Step 2: Compute $\lim_{x \to a} f(x) = \lim_{x \to a} (5x - 3) = 5a - 3$, since the limit of a linear function is just its value at the point.

Step 3: Since $\lim_{x \to a} f(x) = f(a)$ for every real $a$, $f$ is continuous at every real number, including $x = 0$, $x = -3$, and $x = 5$.

The idea of continuity at a point is simple: as you zoom in on the graph at that point, there should be no break, jump, or hole. Formally, for a function $f$ to be continuous at $x = a$, three things must hold:

1. $f(a)$ is defined.
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

For a linear function like $f(x) = 5x - 3$, the graph is a straight line with no breaks anywhere. So we already know it's continuous at every real number. But the problem asks us to prove it at three specific points, which means we show the limit condition holds at each.

Let's do it step by step for each point.

1. At $x = 0$ First, $f(0) = 5(0) - 3 = -3$, so the function is defined. Now compute the limit as $x$ approaches 0:

Since $\lim_{x \to 0} f(x) = -3 = f(0)$, the function is continuous at $x = 0$.

2. At $x = -3$ $f(-3) = 5(-3) - 3 = -15 - 3 = -18$. The limit:

Again, limit equals function value, so continuity holds at $x = -3$.

3. At $x = 5$ $f(5) = 5(5) - 3 = 25 - 3 = 22$. The limit:

So continuity is satisfied at $x = 5$ as well.

Thus, we have proven that $f(x) = 5x - 3$ is continuous at all three points.

Method: Proving Continuity of a Polynomial at One or More Points Using the Polynomial-Continuity Shortcut

Use this whenever asked to "prove" or "show" that a given polynomial is continuous at one or more named points.

Steps

Step 1: Recall the underlying theorem

Every polynomial function is continuous at every real number, because it is built entirely from sums and scalar multiples of power functions $x^n$, each of which is continuous everywhere, and finite sums/scalar multiples of continuous functions are themselves continuous.

Step 2: Compute $f(a)$ directly for each point asked about

Simply substitute the given $x$-value into the polynomial's formula.

Step 3: Compute the limit at each point by direct substitution

Because the polynomial-continuity theorem guarantees $\lim_{x\to a} f(x) = f(a)$, this limit can be evaluated the same way as Step 2 — there is no need for algebraic manipulation, factoring, or an $\varepsilon$-$\delta$ argument.

Step 4: Compare and state the conclusion for each point

Since the limit equals the function value at every point checked, conclude continuity at each named point individually — repeat Steps 2–3 for as many points as the question asks about, since "continuous at $x=0$" is a separate claim from "continuous at $x=-3$," even though the underlying reasoning is identical each time.