Q.Show that the function defined by $f(x) = \cos (x^2)$ is a continuous function.

The key idea is that $f(x)$ is a composition of two continuous functions: $g(x) = x^2$ (a polynomial, continuous everywhere) and $h(t) = \cos t$ (a trigonometric function, continuous everywhere). Since the composition of continuous functions is continuous, $f(x) = h(g(x))$ is continuous on $\mathbb{R}$.

Step 1: $g(x) = x^2$ is continuous on $\mathbb{R}$ (polynomials are continuous).

Step 2: $h(t) = \cos t$ is continuous on $\mathbb{R}$ (cosine is continuous). …

The key idea here is that continuity is preserved under composition. If you have two functions that are each continuous at the relevant points, then their composition is also continuous. This is one of the most powerful shortcuts in analysis — it saves you from having to wrestle with $\epsilon$-$\delta$ proofs every time.

Let’s break it down.

1. Identify the structure of $f$.

   Notice that $f(x) = \cos(x^2)$ can be seen as: first square the input $x$, then take the cosine of the result. So define:

   • $g(x) = x^2$, which maps $\mathbb{R}$ to $[0, \infty)$.
   • $h(u) = \cos u$, which maps $\mathbb{R}$ to $[-1, 1]$. Then $f(x) = h(g(x))$.

2. Check continuity of the inner function $g$.

   $g(x) = x^2$ is a polynomial. Every polynomial is continuous at every real number. So $g$ is continuous on $\mathbb{R}$.

3. Check continuity of the outer function $h$.

   $h(u) = \cos u$ is a standard trigonometric function. From the definition of cosine (or from its graph), we know $\cos u$ is continuous for all real $u$. So $h$ is continuous on $\mathbb{R}$.

4. Apply the Composition Theorem.

   The theorem states: If $g$ is continuous at $x = a$ and $h$ is continuous at $g(a)$, then the composite function $h \circ g$ is continuous at $x = a$.

   Since $g$ is continuous everywhere and $h$ is continuous everywhere, the composition is continuous at every $a \in \mathbb{R}$. …

Method: Proving Continuity by Recognising a Composition

Many show that this function is continuous problems are not proved from first principles at all — they are proved by recognising the function as a composition of two functions you already know are continuous.

Steps

Step 1: Split the function into an inner and an outer function

Write $f(x) = h(g(x))$ where $g$ is computed first, applied directly to $x$, and $h$ is applied to that result.

Step 2: Confirm the inner function $g$ is continuous on its full domain

Most inner functions in this chapter are polynomials, which are continuous everywhere — state this explicitly rather than assuming it.

Step 3: Confirm the outer function $h$ is continuous on the range that $g$ actually produces …

Common Mistakes

Mistake 1: Believing rapid oscillation as $x$ grows breaks continuity

Why it's wrong: continuity is a purely local property — it only asks what happens as $x$ approaches one specific point, never how the function behaves far away or how many times it oscillates overall. $\cos(x^2)$ oscillates faster and faster as $|x|$ grows, but at any fixed point the composition theorem still guarantees a smooth approach. Correct …