Definition:Continuous Real Function/Closed Interval

Definition

Let $f$ be a real function defined on a closed interval $\closedint a b$.

$f$ is continuous on $\closedint a b$ if and only if it is:

$(1): \quad$ continuous at every point of the open interval $\openint a b$

$(2): \quad$ continuous on the right at $a$

$(3): \quad$ continuous on the left at $b$.

That is, if $f$ is to be continuous over the whole of a closed interval, it needs to be continuous at the end points.

Because we only have "access" to the function on one side of each end point, all we can do is insist on continuity on the side of the end points on which the function is defined.

Examples

Example: $\dfrac 1 {1 + e^{1 / x} }$ on $\closedint 0 1$

Consider the real function $f$ defined as:

$f := \begin {cases} \dfrac 1 {1 + e^{1 / x} } & : x \ne 0 \\ 0 & : x = 0 \end {cases}$

Then $f$ is continuous on the closed interval $\closedint 0 1$.

Also see

• Definition:Continuous Real Function on Open Interval
• Definition:Continuous Real Function on Subset

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 9.1$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): continuous function