I'm struggling to understand and how to approach this question, if you could give me a hint about how to answer it I would appreciate that.
So here's the question:
Show, by fixing the value of $F(0)$, that the following function $F(t)$ of a single variable $t$ extends to a continuous function on $R$ (real numbers).
$$F(t)=\frac{e^t - 1}t$$
Cheers
A quotient of two continuous functions is continuous within its domain, and the only points not in the domain are those where the denominator is $0$. It's continuous at $0$ if $\displaystyle\lim_{t\to0} F(t)=F(0)$. So you just need to find that limit and then write $$ F(t)=\begin{cases} (e^t-1)/t & \text{if }t\ne 0, \\ \text{that limit} & \text{if }t=0. \end{cases}$$