diff --git a/frontend/src/utils/latex.test.ts b/frontend/src/utils/latex.test.ts
new file mode 100644
index 0000000..008f8ef
--- /dev/null
+++ b/frontend/src/utils/latex.test.ts
@@ -0,0 +1,72 @@
+import { describe, it, expect } from 'vitest'
+import { escapeHtml, renderLatex, highlightCode } from './latex'
+
+describe('escapeHtml', () => {
+  it('escapes all 5 HTML entities', () => {
+    expect(escapeHtml('&')).toBe('&')
+    expect(escapeHtml('<')).toBe('&lt;')
+    expect(escapeHtml('>')).toBe('&gt;')
+    expect(escapeHtml('"')).toBe('&quot;')
+    expect(escapeHtml("'")).toBe('&#39;')
+  })
+
+  it('escapes multiple entities in a single string', () => {
+    expect(escapeHtml('<b class="x">it\'s &amp;</b>')).toBe(
+      '&lt;b class=&quot;x&quot;&gt;it&#39;s &amp;amp;&lt;/b&gt;'
+    )
+  })
+})
+
+describe('renderLatex', () => {
+  it('returns empty string for empty input', () => {
+    expect(renderLatex('')).toBe('')
+  })
+
+  it('HTML-escapes plain text with dangerous characters', () => {
+    const result = renderLatex('<img src=x onerror=alert(1)>')
+    expect(result).not.toContain('<img')
+    expect(result).toContain('&lt;img')
+    expect(result).toContain('&gt;')
+  })
+
+  it('renders inline math $x^2$ and output contains katex', () => {
+    const result = renderLatex('$x^2$')
+    expect(result.toLowerCase()).toContain('katex')
+  })
+
+  it('renders display math $$x^2$$ and wraps in katex-display-wrap span', () => {
+    const result = renderLatex('$$x^2$$')
+    expect(result).toContain('katex-display-wrap')
+    expect(result.toLowerCase()).toContain('katex')
+  })
+
+  it('HTML-escapes plain text portions alongside math', () => {
+    const result = renderLatex('<b>bold</b> and $x^2$')
+    // Plain text portion must be escaped
+    expect(result).toContain('&lt;b&gt;bold&lt;/b&gt;')
+    // Math portion must still be rendered
+    expect(result.toLowerCase()).toContain('katex')
+  })
+})
+
+describe('highlightCode', () => {
+  it('wraps keywords in cm-keyword spans', () => {
+    const result = highlightCode('const x = 1', 'js')
+    expect(result).toContain('<span class="cm-keyword">const</span>')
+  })
+
+  it('wraps comments in cm-comment spans', () => {
+    const result = highlightCode('// a comment', 'js')
+    expect(result).toContain('cm-comment')
+  })
+
+  it('wraps string literals in cm-string spans', () => {
+    const result = highlightCode('"hello"', 'js')
+    expect(result).toContain('cm-string')
+  })
+
+  it('wraps numbers in cm-number spans', () => {
+    const result = highlightCode('return 42', 'js')
+    expect(result).toContain('<span class="cm-number">42</span>')
+  })
+})
diff --git a/frontend/src/utils/latex.ts b/frontend/src/utils/latex.ts
index a047d86..54d033c 100644
--- a/frontend/src/utils/latex.ts
+++ b/frontend/src/utils/latex.ts
@@ -11,23 +11,33 @@ export const escapeHtml = (s: string): string =>
 export const renderLatex = (text: string): string => {
   if (!text) return ''
   let rest = String(text)
-  // display math: $$...$$
+  const mathPlaceholders: string[] = []
+
+  // Replace math regions with null-byte placeholders before HTML-escaping
   rest = rest.replace(/\$\$([\s\S]+?)\$\$/g, (_, tex: string) => {
+    const idx = mathPlaceholders.length
     try {
-      return `<span class="katex-display-wrap">${katex.renderToString(tex, { displayMode: true, throwOnError: false })}</span>`
+      mathPlaceholders.push(`<span class="katex-display-wrap">${katex.renderToString(tex, { displayMode: true, throwOnError: false })}</span>`)
     } catch {
-      return `<code>${escapeHtml(tex)}</code>`
+      mathPlaceholders.push(`<code>${escapeHtml(tex)}</code>`)
     }
+    return `\x00MATH${idx}\x00`
   })
-  // inline math: $...$
   rest = rest.replace(/\$([^$\n]+?)\$/g, (_, tex: string) => {
+    const idx = mathPlaceholders.length
     try {
-      return katex.renderToString(tex, { displayMode: false, throwOnError: false })
+      mathPlaceholders.push(katex.renderToString(tex, { displayMode: false, throwOnError: false }))
     } catch {
-      return `<code>${escapeHtml(tex)}</code>`
+      mathPlaceholders.push(`<code>${escapeHtml(tex)}</code>`)
     }
+    return `\x00MATH${idx}\x00`
   })
-  return rest
+
+  // Escape plain text (null bytes survive escapeHtml since it only targets HTML chars)
+  rest = escapeHtml(rest)
+
+  // Restore rendered math
+  return rest.replace(/\x00MATH(\d+)\x00/g, (_, i) => mathPlaceholders[Number(i)])
 }
 
 export const highlightCode = (code: string, _lang: string): string => {