Example: Examples of Functions Continuous at a Single Point

(related to Definition: Continuous Functions at Single Real Numbers)

The following are examples of real functions, which are continuous at single points. They exhibit the main ideas of the definition of continuity of a function at a single point:

Example 1:

The function $f:\mathbb R\to \mathbb R$, $f(x):=2x- 3$ is continuous at a point $x=5$.

Proof: * Let $f(x)=2x- 3.$ * Take any $\epsilon > 0$ and set $\delta:=\epsilon/2.$ * Select $x$ such that $0 < |x- 5| < \delta = \epsilon/2.$ * Then $\delta > |x- 5|$ implies $\epsilon > 2|x- 5| = |2x-10| = |(2x- 3)- 7| = |f(x)- 7|.$ * Because $\epsilon$ might have been chosen arbitrarily small, it follows that the limit $$\lim_{x\to 5}f(x)=\lim_{x\to 5}2x- 3=7$$ exists (and equals $7$). * Thus, $f$ is continuous at the point $x=5.$

Example 2

The function $f:\mathbb R\to \mathbb R$, $f(x):=\frac{x+2}{x^2+3x+2}$ is continuous at a point $x=-2$. Please note that the function is not defined for this value of $x$ (division by zero!), even though its limit, as $x$ approaches $-2$, exists.


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