```markdown
### Extended-value extension

The slide discusses the concept of the extended-value extension \(\tilde{f}\) of a function \(f\), defined as:

- \(\tilde{f}(x) = f(x)\), for \(x \in \text{dom } f\)
- \(\tilde{f}(x) = \infty\), for \(x \notin \text{dom } f\)

This extension is useful for simplifying notation. For instance, the condition:

\[ 0 \leq \theta \leq 1 \implies \tilde{f}(\theta x + (1-\theta)y) \leq \theta \tilde{f}(x) + (1-\theta) \tilde{f}(y) \]

(as an inequality in \(\mathbb{R} \cup \{\infty\}\)), is equivalent to the two conditions:

- \(\text{dom } f\) is convex
- For \(x, y \in \text{dom } f\),

\[ 0 \leq \theta \leq 1 \implies f(\theta x + (1-\theta)y) \leq \theta f(x) + (1-\theta) f(y) \]

*Source: A. d'Aspremont. Convex Optimization M2.*
```