Power Function on Strictly Positive Base is Convex

Theorem
Let $a \in \R$ be a real number such that $a > 1$.

Let $f: \R \to \R$ be the real function defined as:
 * $f \left({ x }\right) = a^x$

where $a^x$ denotes $a$ to the power of $x$.

Then $f$ is convex.

Proof
Let $x, y \in \R$.

Note that, from Power of Positive Real Number is Positive: Real Number:
 * $\forall t \in \R : a^t > 0$.

So:

Hence $a^x$ is midpoint-convex.

Further, from Power Function on Strictly Positive Base is Continuous: Real Power, $a^x$ is continuous.

Thus, from Continuous Midpoint-Convex Function is Convex, $a^x$ is convex.