Smooth Functions
Definition
If a function $f$ is infinitely differentiable, then $f$ is called a smooth function.
If a function $f$ is differentiable and $f^{\prime}$ is continuous, then $f$ is called a smooth function.
Explanation
In analysis and functional analysis, the term smooth likely refers to the first definition. The phrase ‘infinitely differentiable’ may seem ambiguous, but it can be understood as follows: $$ \text{For any natural number $n$, the $n$th derivative of $f$ exists.} $$ An infinitely differentiable function is one that can answer “damn yeah” to the question “Can you be differentiated $n$ times?” for any natural number $n$. Especially when discussing function spaces, a smooth function refers to an element of the $C^{\infty}$ space.
Meanwhile, in calculus and geometry, the term smooth likely refers to the second definition. Here, there is not necessarily a need for the function to be infinitely differentiable, especially since calculus does not aim to define and deal with infinitely differentiable as a concept. Therefore, a function is considered smooth if it is as much as $C^{1}$. It shares the same definition as ‘continuously differentiable’.