Reference request: A book to explain the definition $\lim_{n \to \infty} \frac{f(t)-f(t_n)}{t-t_n}$ showing a function is continuous. I have a function $f: I \to \mathbb{R}$ where I is an interval in $\mathbb{R}$.
I have two sequences $t$ and $t_n$. 
I am told that if $$\lim_{n \to \infty} \frac{f(t)-f(t_n)}{t-t_n}$$  exists, then the function $f$ is continuous. 
Is there a book which explains this definition?
In the books I have seen such as Walter Rudin "principles of mathematics", the differentiability of a function is only examined at a point rather than the whole function. 
The context in which I am looking at it is that I am using that definition to prove that a certain function is not differentiable. 
 A: I suppose that you don't have two sequences; rather, fixing $t$ in $I$, we take $t_n$ to be an arbitrary sequence converging to $t$ (and for technical reasons $t_n \neq t$ for all $n$). In this case, the existence of the given limit is equivalent to the existence of the derivative $f'$ at the point $t$. If the limit exists, then we see \begin{align*} \lim_{n\to\infty} \lvert f(t) - f(t_n) \rvert &= \lim_{n\to\infty} \lvert t -t_n \rvert \frac{\lvert f(t) - f(t_n) \rvert}{\lvert t - t_n \rvert}\\
&= \left( \lim_{n\to\infty} \lvert t -t_n \rvert\right) \left(\lim_{n\to \infty} \frac{\lvert f(t) - f(t_n) \rvert}{\lvert t - t_n \rvert}\right) = 0 \cdot \lvert f'(t) \rvert = 0.
\end{align*} Thus whenever $t_n \to t$, we have $f(t_n) \to f(t)$ which implies continuity of $f$ at $t$ by the sequential criterion theorem. 
This is not a definition of continuity; indeed a function may be continuous and the given limit may fail to exist. However, existence of the limit implies the continuity of $f$.
