# Category:Convex Real Functions

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This category contains results about **Convex Real Functions**.

Definitions specific to this category can be found in **Definitions/Convex Real Functions**.

$f$ is **convex on $I$** if and only if:

- $\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$

## Also see

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Convex Real Functions"

The following 28 pages are in this category, out of 28 total.

### C

- Conditional Jensen's Inequality
- Convex Real Function Composed with Increasing Convex Real Function is Convex
- Convex Real Function is Continuous
- Convex Real Function is Left-Hand and Right-Hand Differentiable
- Convex Real Function is Measurable
- Convex Real Function is Pointwise Supremum of Affine Functions

### E

### I

- Inverse of Strictly Decreasing Convex Real Function is Convex
- Inverse of Strictly Decreasing Strictly Convex Real Function is Strictly Convex
- Inverse of Strictly Increasing Concave Real Function is Convex
- Inverse of Strictly Increasing Convex Real Function is Concave
- Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex
- Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave