- Inicio
- Continuity
- Continuity of a function at a point
- Ejercicios
Continuity of a function at a point
Let's look at the continuity of the function
$$\displaystyle f(x)=\left\{\begin{array}{rcl} x^2+2 & \mbox{ if } & x<1 \\ 3x & \mbox{ if } & x \geq 1\end{array}\right.$$
The functions that define $f(x)$ are continuous since they are polynomial, therefore they can only be discontinuous if they do not connect when $x=1$.
$$\displaystyle \begin{array} {l} \lim_{x \to 1^-}f(x)=\lim_{x \to 1} (x^2+2)= 1^2+2=3 \\ \lim_{x \to 1^+}f(x)=\lim_{x \to 1} (3x)= 3 \\ f(1)=3\cdot1=3 \end{array}$$ As the lateral limits coincide with the value of the function, the function is continuous.
The function is continuous when $x=1$ and in all $\mathbb{R}$.
Let's look at the continuity of the function
$$\displaystyle f(x)=\left\{\begin{array}{rcl} x-3 & \mbox{ if } & x\neq3 \\ 0 & \mbox{ if } & x=3 \end{array}\right.$$
The functions that define $f(x)$ are continuous since they are polynomial, therefore they can only be discontinuous if the two functions do not connect when $x=3$.
$$\displaystyle \begin{array} {l} \lim_{x \to 3^-}f(x)=\lim_{x \to 3} (x-3)= 0 \\ \lim_{x \to 3^+}f(x)=\lim_{x \to 3} (x-3)= 0 \\ f(3)=0 \end{array}$$ As the lateral limits coincide with the value of the function, the function is continuous.
The function is continuous when $x=3$ and in all $\mathbb{R}$.