Answer
(a) $f$ is continuous at $a$ by definition means that $$\lim\limits_{x\to a}f(x)=f(a)$$
(b) $f$ is continuous on the interval $(-\infty,\infty)$ means $f$ is continuous at any point $x\in R$.
The graph of such a function stretches from $-\infty$ to $\infty$ and is a continuous line from $-\infty$ to $\infty$.
Work Step by Step
(a) $f$ is continuous at $a$ by definition means that $$\lim\limits_{x\to a}f(x)=f(a)$$
This also means that $f(a)$ is defined and $\lim\limits_{x\to a}f(x)$ exists.
In the graph, $f$ is continuous at $a$ is shown as a unbroken line at point $a$
We can intuitively think that since the line is continuous at point $x=a$, as $x$ approaches $a$, the graph of $f$ would approach continuously towards $f(a)$. So $\lim\limits_{x\to a}f(x)=f(a)$
(b) $f$ is continuous on the interval $(-\infty,\infty)$ means $f$ is continuous at any point $x\in R$.
The graph of such a function stretches from $-\infty$ to $\infty$ and is a continuous line from $-\infty$ to $\infty$.
According to definition, $f$ is continuous on an interval if it is continuous at any number in the interval. That means if $f$ is continuous on the interval $(-\infty,\infty)$, it is continuous at any number on $(-\infty,\infty)$, which is the whole set of rational numbers $R$.
