Show that the vector function 𝐫 defined by 𝐫(t)=f(t) 𝐢+g(t) 𝐣+h(t) 𝐤 is continuous at t=t0 if an

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Show that the vector function 𝐫 defined by 𝐫(t)=f(t) 𝐢+g(t)...

Show that the vector function $\mathbf{r}$ defined by $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ is continuous at $t=t_{0}$ if and only if $f, g,$ and $h$ are continuous at $t_{0}$.

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Key Concepts

Continuity of a Vector Function

In vector calculus, a vector function is said to be continuous at a point if and only if the limit of the function as the variable approaches that point equals the function's value at that point. This definition extends the idea of continuity from scalar functions to vector functions by using the concept of limits in a multi-dimensional context.

Component-wise Continuity

A key insight into the continuity of a vector function is that it can be analyzed by examining its individual scalar component functions. If each component function (typically corresponding to the i, j, and k directions in 3-dimensional space) is continuous at a given point, then the vector function itself is continuous at that point. Conversely, if the vector function is continuous, then each of its component functions must be continuous.

Limit of a Vector Function

The limit of a vector function as its variable approaches a specific value exists if and only if the limits of its individual component functions exist and are equal to the corresponding components of the vector function's value at that point. This concept is fundamental in proving continuity as it allows the examination of the behavior of complex vector functions through simpler scalar limits.

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