Question

If c∈Vn, show that the function f given by f(x)=c.x is continuous on R^n.

Answer 1

$$Solution$$

The function $f: \rm I\!R^2 \to \rm I\!R$ defined by $f(x)=c \cdot x$ for a fixed $c$.

I'll show that $f$ is continuous at a point $x_0 \in \rm I\!R^n$ . Given any $e >0$, we need to show that there exists $\delta >0$ such that $\| x-x_0\|<\delta \implies |f(x)-f(x_0)|<e$

Now consider $|f(x)-f(x_0)|=|c \cdot x-c \cdot x_0|=|c \cdot (x-x_0)| \leq \|c\|\ \|x-x_0\|$

Choose $\delta= \frac{\|c\|}{e} \implies |f(x)-f(x_0)|\leq \|c\| \|x-x_0\|<\|c\| \delta=e$