Continuity

Continuity for real valued functions

A function $f:X\to Y$ on $\mathbb{R}$ is continuous at $x_0$ if for every sequence $\{a_n\}\subset X$ such that $a_n\to x_0$, we have $f(a_n)\to f(x_0)$. Equivalently, $f$ is continuous at $x_0$ if for all $\epsilon >0$ there exists $\delta >0$ such that $|x-x_0|<\delta ⟹|f(x)-f(x_0)|<\epsilon$. We say $f$ is continuous on $X$ if it is continuous at every $x_0\in X$.

Example: Polynomial functions are continuous on $\mathbb{R}$.

Proposition

• Suppose $g$ is continuous at a and $f$ is continuous at $g(a)$ then $f(g(x))$ is continuous at a.
• If $f$ is injective and continuous then $f^{-1}$ is continuous.

Intermediate Value Theorem

If $f:[a,b]\to \mathbb{R}$ is continuous, then for any real $\gamma$ such that $\min (f(a),f(b))\le \gamma \le \max (f(a),f(b))$, there exists $c\in [a,b]$ such that $f(c)=\gamma$.

Extreme Value Theorem

If $f:[a,b]\to \mathbb{R}$ is continuous, then $f$ is bounded, and attains its bounds.